JEE Main & Advanced Mathematics Differentiation Question Bank

done Derivative at a point Standard Differentiation Total Questions - 183

• question_answer1)
  \[\frac{d}{dx}(\log \tan x)=\]                                           [MNR 1986]
  A)
             \[2\sec 2x\] done clear
  B)
             \[2\,\text{cosec }2x\] done clear
  C)
             \[\sec 2x\] done clear
  D)
             \[\text{cosec}\,2x\] done clear
  View Solution play_arrow

• question_answer2)
  \[\frac{d}{dx}\log (\log x)\]=                                   [IIT 1985]
  A)
             \[\frac{x}{\log x}\] done clear
  B)
             \[\frac{\log x}{x}\] done clear
  C)
             \[{{(x\log x)}^{-1}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer3)
  \[\frac{d}{dx}{{\left( \sqrt{x}+\frac{1}{\sqrt{x}} \right)}^{2}}=\]                         [AI CBSE 1980]
  A)
             \[1-\frac{1}{{{x}^{2}}}\] done clear
  B)
             \[1+\frac{1}{{{x}^{2}}}\] done clear
  C)
             \[1-\frac{1}{2x}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer4)
  If \[y=x+\frac{1}{x}\], then
  A)
             \[{{x}^{2}}\frac{dy}{dx}+xy=0\] done clear
  B)
             \[{{x}^{2}}\frac{dy}{dx}+xy+2=0\] done clear
  C)
             \[{{x}^{2}}\frac{dy}{dx}-xy+2=0\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer5)
  \[\frac{d}{dx}\left( \frac{1}{{{x}^{4}}\sec x} \right)=\]     
  A)
             \[\frac{x\sin x+4\cos x}{{{x}^{5}}}\] done clear
  B)
             \[\frac{-(x\sin x+4\cos x)}{{{x}^{5}}}\] done clear
  C)
                     \[\frac{4\cos x-x\sin x}{{{x}^{5}}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer6)
  If \[y=1+x+\frac{{{x}^{2}}}{2\,!}+\frac{{{x}^{3}}}{3\,!}+.....+\frac{{{x}^{n}}}{n\,!}\], then \[\frac{dy}{dx}=\]
  A)
             y done clear
  B)
             \[y+\frac{{{x}^{n}}}{n!}\] done clear
  C)
             \[y-\frac{{{x}^{n}}}{n!}\] done clear
  D)
             \[y-1-\frac{{{x}^{n}}}{n!}\] done clear
  View Solution play_arrow

• question_answer7)
  If \[y=1+x+\frac{{{x}^{2}}}{2!}+\frac{{{x}^{3}}}{3!}+.....\infty ,\]then \[\frac{dy}{dx}=\] [Karnataka CET 1999]
  A)
             y done clear
  B)
             \[y-1\] done clear
  C)
             \[y+1\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer8)
  If \[y=\frac{1}{a-z},\]then \[\frac{dz}{dy}=\]
  A)
             \[{{(z-a)}^{2}}\] done clear
  B)
             \[-{{(z-a)}^{2}}\] done clear
  C)
             \[{{(z+a)}^{2}}\] done clear
  D)
             \[-{{(z+a)}^{2}}\] done clear
  View Solution play_arrow

• question_answer9)
  If \[y=x\sin x,\]then                      
  A)
             \[\frac{1}{y}\frac{dy}{dx}=\frac{1}{x}+\cot x\] done clear
  B)
             \[\frac{dy}{dx}=\frac{1}{x}+\cot x\] done clear
  C)
             \[\frac{1}{y}\frac{dy}{dx}=\frac{1}{x}-\cot x\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer10)
  \[\frac{d}{dx}({{x}^{2}}{{e}^{x}}\sin x)=\]
  A)
             \[x\,{{e}^{x}}(2\sin x+x\sin x+x\cos x)\] done clear
  B)
             \[x\,{{e}^{x}}(2\sin x+x\sin x-\cos x)\] done clear
  C)
             \[x\,{{e}^{x}}(2\sin x+x\sin x+\cos x)\]                                 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer11)
  \[\frac{d}{dx}\left( {{\tan }^{-1}}\frac{\cos x}{1+\sin x} \right)=\] [AISSE 1984, 85; MNR 1983; RPET 1997]
  A)
             \[-\frac{1}{2}\] done clear
  B)
             \[\frac{1}{2}\] done clear
  C)
             \[-1\] done clear
  D)
             1 done clear
  View Solution play_arrow

• question_answer12)
  \[\frac{d}{dx}[\cos {{(1-{{x}^{2}})}^{2}}]\]=                [AISSE 1981; AI CBSE 1979]
  A)
             \[-2x(1-{{x}^{2}})\sin {{(1-{{x}^{2}})}^{2}}\] done clear
  B)
   \[-4x(1-{{x}^{2}})\sin {{(1-{{x}^{2}})}^{2}}\] done clear
  C)
             \[4x(1-{{x}^{2}})\sin {{(1-{{x}^{2}})}^{2}}\] done clear
  D)
             \[-2(1-{{x}^{2}})\sin {{(1-{{x}^{2}})}^{2}}\] done clear
  View Solution play_arrow

• question_answer13)
  \[\frac{d}{dx}\left( {{x}^{2}}\sin \frac{1}{x} \right)=\]
  A)
             \[\cos \,\left( \frac{1}{x} \right)+2x\sin \left( \frac{1}{x} \right)\] done clear
  B)
             \[2x\sin \left( \frac{1}{x} \right)-\cos \left( \frac{1}{x} \right)\] done clear
  C)
             \[\cos \left( \frac{1}{x} \right)-2x\sin \left( \frac{1}{x} \right)\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer14)
  If \[y=\cos (\sin {{x}^{2}}),\]then at \[x=\sqrt{\frac{\pi }{2}},\frac{dy}{dx}\]=
  A)
             ? 2 done clear
  B)
             2 done clear
  C)
             \[-2\sqrt{\frac{\pi }{2}}\] done clear
  D)
           0 done clear
  View Solution play_arrow

• question_answer15)
  If \[y={{\sin }^{-1}}(x\sqrt{1-x}+\sqrt{x}\sqrt{1-{{x}^{2}})},\]then \[\frac{dy}{dx}=\] [Roorkee 1981; MP PET 2004]
  A)
             \[\frac{-2x}{\sqrt{1-{{x}^{2}}}}+\frac{1}{2\sqrt{x-{{x}^{2}}}}\] done clear
  B)
             \[\frac{-1}{\sqrt{1-{{x}^{2}}}}-\frac{1}{2\sqrt{x-{{x}^{2}}}}\] done clear
  C)
             \[\frac{1}{\sqrt{1-{{x}^{2}}}}+\frac{1}{2\sqrt{x-{{x}^{2}}}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer16)
  \[\frac{d}{dx}\log |x|\text{ }=......,(x\ne 0)\]                     
  A)
             \[\frac{1}{x}\] done clear
  B)
             \[-\frac{1}{x}\] done clear
  C)
             x done clear
  D)
             \[-x\] done clear
  View Solution play_arrow

• question_answer17)
  If \[y=a\sin x+b\cos x,\]then \[{{y}^{2}}+{{\left( \frac{dy}{dx} \right)}^{2}}\]is a
  A)
             Function of x done clear
  B)
             Function of y done clear
  C)
             Function of x and y done clear
  D)
             Constant done clear
  View Solution play_arrow

• question_answer18)
  \[f(x)={{x}^{2}}-3x\], then the points at which \[f(x)=f'(x)\]are
  A)
             1, 3 done clear
  B)
             1, ? 3 done clear
  C)
             ? 1, 3 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer19)
  If \[f(x)=mx+c,f(0)=f'(0)=1\]then \[f(2)=\]
  A)
             1 done clear
  B)
             2 done clear
  C)
             3 done clear
  D)
             ? 3 done clear
  View Solution play_arrow

• question_answer20)
  If \[y=3{{x}^{5}}+4{{x}^{4}}+2x+3\], then
  A)
             \[{{y}_{4}}=0\] done clear
  B)
             \[{{y}_{5}}=0\] done clear
  C)
             \[{{y}_{6}}=0\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer21)
  If \[y=x\text{ }\left[ \left( \cos \frac{x}{2}+\sin \frac{x}{2} \right)\text{ }\left( \cos \frac{x}{2}-\sin \frac{x}{2} \right)+\sin x \right]+\frac{1}{2\sqrt{x}}\], then \[\frac{dy}{dx}=\]
  A)
             \[(1+x)\cos x+(1-x)\sin x-\frac{1}{4x\sqrt{x}}\] done clear
  B)
             \[(1-x)\cos x+(1+x)\sin x+\frac{1}{4x\sqrt{x}}\] done clear
  C)
             \[(1+x)\cos x+(1+x)\sin x-\frac{1}{4x\sqrt{x}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer22)
  The differential coefficient of \[{{a}^{x}}+\log x.\sin x\]is
  A)
             \[{{a}^{x}}{{\log }_{e}}a+\frac{\sin x}{x}+\log x.\cos x\] done clear
  B)
             \[{{a}^{x}}+\frac{\sin x}{x}+\cos x.\log x\] done clear
  C)
             \[{{a}^{x}}\log a+\frac{\cos x}{x}+\sin x.\log x.\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer23)
  \[\frac{d}{dx}{{\tan }^{-1}}\left( \frac{ax-b}{bx+a} \right)=\]
  A)
             \[\frac{1}{1+{{x}^{2}}}-\frac{{{a}^{2}}}{{{a}^{2}}+{{b}^{2}}}\] done clear
  B)
             \[\frac{-1}{1+{{x}^{2}}}-\frac{{{a}^{2}}}{{{a}^{2}}+{{b}^{2}}}\] done clear
  C)
             \[\frac{1}{1+{{x}^{2}}}+\frac{{{a}^{2}}}{{{a}^{2}}+{{b}^{2}}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer24)
  \[\frac{d}{dx}\left( {{\tan }^{-1}}\sqrt{\frac{1+\cos \frac{x}{2}}{1-\cos \frac{x}{2}}} \right)\]is equal to [MP PET 2004]
  A)
             \[-\frac{1}{4}\] done clear
  B)
             \[\frac{1}{2}\] done clear
  C)
             \[-\frac{1}{2}\] done clear
  D)
             \[\frac{1}{4}\] done clear
  View Solution play_arrow

• question_answer25)
  \[\frac{d}{dx}\sqrt{\frac{1-\sin 2x}{1+\sin 2x}}=\]     [AISSE 1985; DSSE 1986]
  A)
             \[{{\sec }^{2}}x\] done clear
  B)
  \[-{{\sec }^{2}}\left( \frac{\pi }{4}-x \right)\] done clear
  C)
             \[{{\sec }^{2}}\left( \frac{\pi }{4}+x \right)\] done clear
  D)
             \[{{\sec }^{2}}\left( \frac{\pi }{4}-x \right)\] done clear
  View Solution play_arrow

• question_answer26)
  If \[y=\sqrt{(1-x)(1+x)}\], then
  A)
             \[(1-{{x}^{2}})\frac{dy}{dx}-xy=0\] done clear
  B)
             \[(1-{{x}^{2}})\frac{dy}{dx}+xy=0\] done clear
  C)
             \[(1-{{x}^{2}})\frac{dy}{dx}-2xy=0\] done clear
  D)
   \[(1-{{x}^{2}})\frac{dy}{dx}+2xy=0\] done clear
  View Solution play_arrow

• question_answer27)
  \[\frac{d}{dx}\left( \frac{{{\cot }^{2}}x-1}{{{\cot }^{2}}x+1} \right)=\]
  A)
             \[-\sin 2x\] done clear
  B)
             \[2\sin 2x\] done clear
  C)
             \[2\cos 2x\] done clear
  D)
             \[-2\sin 2x\] done clear
  View Solution play_arrow

• question_answer28)
  If \[f(x)=x{{\tan }^{-1}}x\], then \[f'(1)\]=                                                    [IIT 1979]
  A)
             \[1+\frac{\pi }{4}\] done clear
  B)
             \[\frac{1}{2}+\frac{\pi }{4}\] done clear
  C)
             \[\frac{1}{2}-\frac{\pi }{4}\] done clear
  D)
             2 done clear
  View Solution play_arrow

• question_answer29)
  If \[y={{\log }_{10}}x+{{\log }_{x}}10+{{\log }_{x}}x+{{\log }_{10}}10,\]then \[\frac{dy}{dx}=\]
  A)
             \[\frac{1}{x{{\log }_{e}}10}-\frac{{{\log }_{e}}10}{x{{({{\log }_{e}}x)}^{2}}}\] done clear
  B)
             \[\frac{1}{x{{\log }_{e}}10}-\frac{1}{x{{\log }_{10}}e}\] done clear
  C)
             \[\frac{1}{x{{\log }_{e}}10}-\frac{{{\log }_{e}}10}{x{{({{\log }_{e}}x)}^{2}}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer30)
  If \[y=b\cos \log {{\left( \frac{x}{n} \right)}^{n}}\], then \[\frac{dy}{dx}=\]
  A)
             \[-n\,\,b\sin \log {{\left( \frac{x}{n} \right)}^{n}}\] done clear
  B)
             \[n\,\,b\sin \log {{\left( \frac{x}{n} \right)}^{n}}\] done clear
  C)
             \[-n\,b\sin \log {{\left( \frac{x}{n} \right)}^{n}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer31)
  \[\frac{d}{dx}[{{\sin }^{n}}x\cos \,nx]=\]
  A)
             \[n{{\sin }^{n-1}}x\cos (n+1)x\] done clear
  B)
             \[n{{\sin }^{n-1}}x\cos \,nx\] done clear
  C)
             \[n{{\sin }^{n-1}}x\cos (n-1)x\] done clear
  D)
             \[n{{\sin }^{n-1}}x\sin (n+1)x\] done clear
  View Solution play_arrow

• question_answer32)
  If \[f(x)={{\log }_{x}}(\log x),\]then \[f'(x)\]at \[x=e\]is [IIT 1985; RPET 2000; MP PET 2000; Karnataka CET 2002; Pb. CET 2002]
  A)
             e done clear
  B)
             \[\frac{1}{e}\] done clear
  C)
             1 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer33)
  If \[y=\log {{\left( \frac{1+x}{1-x} \right)}^{1/4}}-\frac{1}{2}{{\tan }^{-1}}x,\]then \[\frac{dy}{dx}=\]
  A)
             \[\frac{{{x}^{2}}}{1-{{x}^{4}}}\] done clear
  B)
             \[\frac{2{{x}^{2}}}{1-{{x}^{4}}}\] done clear
  C)
             \[\frac{{{x}^{2}}}{2\,\,(1-{{x}^{4}})}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer34)
  If \[y={{\tan }^{-1}}\frac{4x}{1+5{{x}^{2}}}+{{\tan }^{-1}}\frac{2+3x}{3-2x}\], then \[\frac{dy}{dx}=\]
  A)
             \[\frac{1}{1+25{{x}^{2}}}+\frac{2}{1+{{x}^{2}}}\] done clear
  B)
             \[\frac{5}{1+25{{x}^{2}}}+\frac{2}{1+{{x}^{2}}}\] done clear
  C)
           \[\frac{5}{1+25{{x}^{2}}}\] done clear
  D)
             \[\frac{1}{1+25{{x}^{2}}}\] done clear
  View Solution play_arrow

• question_answer35)
  \[\frac{d}{dx}{{\log }_{7}}({{\log }_{7}}x)\]=
  A)
             \[\frac{1}{x{{\log }_{e}}x}\] done clear
  B)
             \[\frac{{{\log }_{e}}7}{x{{\log }_{e}}x}\] done clear
  C)
           \[\frac{{{\log }_{7}}e}{x{{\log }_{e}}x}\] done clear
  D)
             \[\frac{{{\log }_{7}}e}{x{{\log }_{7}}x}\] done clear
  View Solution play_arrow

• question_answer36)
  If \[f(x)=\sqrt{1+{{\cos }^{2}}({{x}^{2}})}\], then \[f'\left( \frac{\sqrt{\pi }}{2} \right)\] is   [Orissa JEE 2004]
  A)
             \[\sqrt{\pi }/6\] done clear
  B)
             \[-\,\sqrt{(\pi /6)}\] done clear
  C)
             \[1/\sqrt{6}\] done clear
  D)
             \[\pi /\sqrt{6}\] done clear
  View Solution play_arrow

• question_answer37)
  If \[{{x}^{2/3}}+{{y}^{2/3}}={{a}^{2/3}}\], then \[\frac{dy}{dx}=\]
  A)
             \[{{\left( \frac{y}{x} \right)}^{1/3}}\] done clear
  B)
             \[-{{\left( \frac{y}{x} \right)}^{1/3}}\] done clear
  C)
             \[{{\left( \frac{x}{y} \right)}^{1/3}}\] done clear
  D)
             \[-{{\left( \frac{x}{y} \right)}^{1/3}}\] done clear
  View Solution play_arrow

• question_answer38)
  \[\frac{d}{dx}[(1+{{x}^{2}}){{\tan }^{-1}}x]=\]
  A)
             \[x{{\tan }^{-1}}x\] done clear
  B)
             \[2{{\tan }^{-1}}x\] done clear
  C)
             \[2x{{\tan }^{-1}}x+1\] done clear
  D)
             \[x{{\tan }^{-1}}x+1\] done clear
  View Solution play_arrow

• question_answer39)
  If \[y=\log \frac{1+\sqrt{x}}{1-\sqrt{x}},\]then \[\frac{dy}{dx}=\]
  A)
             \[\frac{\sqrt{x}}{1-x}\] done clear
  B)
             \[\frac{1}{\sqrt{x}(1-x)}\] done clear
  C)
             \[\frac{\sqrt{x}}{1+x}\] done clear
  D)
             \[\frac{1}{\sqrt{x}(1+x)}\] done clear
  View Solution play_arrow

• question_answer40)
  \[\frac{d}{dx}{{e}^{x+3\log x}}=\]
  A)
             \[{{e}^{x}}.{{x}^{2}}(x+3)\] done clear
  B)
             \[{{e}^{x}}.x(x+3)\] done clear
  C)
             \[{{e}^{x}}+\frac{3}{x}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer41)
  \[\frac{d}{dx}\sqrt{\frac{1+\cos 2x}{1-\cos 2x}}=\]
  A)
             \[{{\sec }^{2}}x\] done clear
  B)
           \[-\text{cose}{{\text{c}}^{2}}x\] done clear
  C)
             \[2\,{{\sec }^{2}}\frac{x}{2}\] done clear
  D)
             \[-2\text{cose}{{\text{c}}^{2}}\frac{x}{2}\] done clear
  View Solution play_arrow

• question_answer42)
  \[\frac{d}{dx}\log \tan \left( \frac{\pi }{4}+\frac{x}{2} \right)=\]
  A)
             \[\cos \text{ec}\,x\] done clear
  B)
             \[-\cos \text{ec}\,x\] done clear
  C)
             \[\sec x\] done clear
  D)
             \[-\sec x\] done clear
  View Solution play_arrow

• question_answer43)
  \[\frac{d}{dx}\log (\sqrt{x-a}+\sqrt{x-b})=\]
  A)
             \[\frac{1}{2[\sqrt{(x-a)}+\sqrt{(x-b)}]}\] done clear
  B)
           \[\frac{1}{2\sqrt{(x-a)(x-b)}}\] done clear
  C)
             \[\frac{1}{\sqrt{(x-a)(x-b)}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer44)
  \[\frac{d}{dx}{{\tan }^{-1}}(\sec x+\tan x)=\]  [AISSE 1985, 87; DSSE 1982, 84]
  A)
             1 done clear
  B)
             1/2 done clear
  C)
             \[\cos x\] done clear
  D)
             \[\sec x\] done clear
  View Solution play_arrow

• question_answer45)
  \[\frac{d}{dx}{{\cos }^{-1}}\sqrt{\cos x}=\]
  A)
           \[\frac{1}{2}\sqrt{1+\sec x}\] done clear
  B)
             \[\sqrt{1+\sec x}\] done clear
  C)
             \[-\frac{1}{2}\sqrt{1+\sec x}\] done clear
  D)
             \[-\sqrt{1+\sec x}\] done clear
  View Solution play_arrow

• question_answer46)
  \[\frac{d}{dx}({{e}^{x}}\log \sin 2x)=\] [AI CBSE 1985]
  A)
             \[{{e}^{x}}(\log \sin 2x+2\cot 2x)\] done clear
  B)
             \[{{e}^{x}}(\log \cos 2x+2\cot 2x)\] done clear
  C)
             \[{{e}^{x}}(\log \cos 2x+\cot 2x)\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer47)
   \[\frac{d}{dx}{{\tan }^{-1}}\frac{4\sqrt{x}}{1-4x}=\]
  A)
             \[\frac{1}{\sqrt{x}(1+4x)}\] done clear
  B)
             \[\frac{2}{\sqrt{x}(1+4x)}\] done clear
  C)
             \[\frac{4}{\sqrt{x}(1+4x)}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer48)
   If \[y=\sin [\cos (\sin x)],\]then \[dy/dx=\]
  A)
             \[-\cos [\cos (\sin x)]\sin (\cos x).\cos x\] done clear
  B)
             \[-\cos [\cos (\sin x)]\sin (\sin x).\cos x\] done clear
  C)
             \[\cos [\cos (\sin x)]\sin (\cos x).\cos x\] done clear
  D)
             \[\cos [\cos (\sin x)]\sin (\sin x).\cos x\] done clear
  View Solution play_arrow

• question_answer49)
  If \[y={{\sec }^{-1}}\left( \frac{\sqrt{x}+1}{\sqrt{x}-1} \right)+{{\sin }^{-1}}\left( \frac{\sqrt{x}-1}{\sqrt{x}+1} \right)\], then \[\frac{dy}{dx}=\] [UPSEAT 1999; AMU 2002; Kerala (Engg.) 2005]
  A)
             0 done clear
  B)
             \[\frac{1}{\sqrt{x}+1}\] done clear
  C)
             1 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer50)
  \[\frac{d}{dx}{{\sin }^{-1}}(3x-4{{x}^{3}})=\]                                                     [RPET 2003]
  A)
             \[\frac{3}{\sqrt{1-{{x}^{2}}}}\] done clear
  B)
             \[\frac{-3}{\sqrt{1-{{x}^{2}}}}\] done clear
  C)
             \[\frac{1}{\sqrt{1-{{x}^{2}}}}\] done clear
  D)
             \[\frac{-1}{\sqrt{1-{{x}^{2}}}}\] done clear
  View Solution play_arrow

• question_answer51)
  If \[y={{t}^{4/3}}-3{{t}^{-2/3}}\], then \[dy/dt\]=
  A)
             \[\frac{2{{t}^{2}}+3}{3{{t}^{5/3}}}\] done clear
  B)
             \[\frac{2{{t}^{2}}+3}{{{t}^{5/3}}}\] done clear
  C)
             \[\frac{2(2{{t}^{2}}+3)}{{{t}^{5/3}}}\] done clear
  D)
           \[\frac{2(2{{t}^{2}}+3)}{3{{t}^{5/3}}}\] done clear
  View Solution play_arrow

• question_answer52)
  If \[y={{x}^{2}}\log x+\frac{2}{\sqrt{x}},\] then \[\frac{dy}{dx}=\]
  A)
             \[x+2x\log x-\frac{1}{\sqrt{x}}\] done clear
  B)
             \[x+2x\log x-\frac{1}{{{x}^{3/2}}}\] done clear
  C)
             \[x+2x\log x-\frac{2}{{{x}^{3/2}}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer53)
  \[\frac{d}{dx}\left( \frac{{{e}^{x}}}{1+{{x}^{2}}} \right)=\]
  A)
             \[\frac{{{e}^{x}}(1+x)}{{{(1+{{x}^{2}})}^{2}}}\] done clear
  B)
             \[\frac{{{e}^{x}}{{(1-x)}^{2}}}{{{(1+{{x}^{2}})}^{2}}}\] done clear
  C)
             \[\frac{{{e}^{x}}{{(1+x)}^{2}}}{(1+{{x}^{2}})}\]      done clear
  D)
             \[\frac{{{e}^{x}}{{(1-x)}^{2}}}{(1+{{x}^{2}})}\] done clear
  View Solution play_arrow

• question_answer54)
  If \[y=\frac{\tan x+\cot x}{\tan x-\cot x},\]then \[\frac{dy}{dx}=\]
  A)
             \[2\tan 2x\sec 2x\] done clear
  B)
             \[\tan 2x\sec 2x\] done clear
  C)
             \[-\tan 2x\sec 2x\] done clear
  D)
           \[-2\tan 2x\sec 2x\] done clear
  View Solution play_arrow

• question_answer55)
  If \[A=\frac{{{2}^{x}}\cot x}{\sqrt{x}},\]then \[\frac{dA}{dx}=\]
  A)
           \[\frac{{{2}^{x-1}}\left\{ -2x\,\text{cos}\text{e}{{\text{c}}^{2}}x+\cot x.\log \left( \frac{{{4}^{x}}}{e} \right) \right\}}{{{x}^{3/2}}}\] done clear
  B)
             \[\frac{{{2}^{x-1}}\left\{ -2x\cos \text{e}{{\text{c}}^{2}}x+\cot x.\log \left( \frac{{{4}^{x}}}{e} \right) \right\}}{x}\] done clear
  C)
             \[\frac{2x\left\{ -2x\text{cose}{{\text{c}}^{2}}x+\cot x.\log \left( \frac{{{4}^{x}}}{e} \right) \right\}}{{{x}^{\text{3/2}}}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer56)
  \[\frac{d}{dx}\left( \frac{\log x}{\sin x} \right)=\]                               
  A)
             \[\frac{\frac{\sin x}{x}-\log x.\cos x}{\sin x}\] done clear
  B)
             \[\frac{\frac{\sin x}{x}-\log x.\cos x}{{{\sin }^{2}}x}\] done clear
  C)
             \[\frac{\sin x-\log x.\cos x}{{{\sin }^{2}}x}\] done clear
  D)
             \[\frac{\frac{\sin x}{x}-\log x}{{{\sin }^{2}}x}\] done clear
  View Solution play_arrow

• question_answer57)
  If \[y=\frac{\sqrt{{{x}^{2}}+1}+\sqrt{{{x}^{2}}-1}}{\sqrt{{{x}^{2}}+1}-\sqrt{{{x}^{2}}-1}}\], then \[\frac{dy}{dx}=\]
  A)
             \[2x+\frac{2{{x}^{3}}}{\sqrt{{{x}^{4}}-1}}\] done clear
  B)
             \[2x+\frac{{{x}^{3}}}{\sqrt{{{x}^{4}}-1}}\] done clear
  C)
             \[x+\frac{2{{x}^{3}}}{\sqrt{{{x}^{4}}-1}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer58)
  If \[y=\frac{\sqrt{a+x}-\sqrt{a-x}}{\sqrt{a+x}+\sqrt{a-x}}\], then  \[\frac{dy}{dx}=\]             [AISSE 1986]
  A)
           \[\frac{ay}{x\sqrt{{{a}^{2}}-{{x}^{2}}}}\] done clear
  B)
             \[\frac{ay}{\sqrt{{{a}^{2}}-{{x}^{2}}}}\] done clear
  C)
             \[\frac{ay}{x\sqrt{{{x}^{2}}-{{a}^{2}}}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer59)
   If \[y={{(x{{\cot }^{3}}x)}^{3/2}},\]then \[dy/dx=\]
  A)
             \[\frac{3}{2}{{(x{{\cot }^{3}}x)}^{1/2}}[{{\cot }^{3}}x-3x{{\cot }^{2}}x\cos \text{e}{{\text{c}}^{2}}x]\] done clear
  B)
             \[\frac{3}{2}{{(x{{\cot }^{3}}x)}^{1/2}}[{{\cot }^{2}}x-3x{{\cot }^{2}}x\,\text{cose}{{\text{c}}^{2}}x]\] done clear
  C)
             \[\frac{3}{2}{{(x{{\cot }^{3}}x)}^{1/3}}[{{\cot }^{3}}x-3x\,\text{cos}\text{e}{{\text{c}}^{2}}x]\] done clear
  D)
             \[\frac{3}{2}{{(x{{\cot }^{3}}x)}^{3/2}}[{{\cot }^{3}}x-3x\,\text{cos}\text{e}{{\text{c}}^{2}}x]\] done clear
  View Solution play_arrow

• question_answer60)
  \[\frac{d}{dx}\{\cos (\sin {{x}^{2}})\}=\]                                                                   [DSSE 1979]
  A)
             \[\sin (\sin {{x}^{2}}).\cos {{x}^{2}}.2x\] done clear
  B)
             \[-\sin (\sin {{x}^{2}}).\cos {{x}^{2}}.2x\] done clear
  C)
             \[-\sin (\sin {{x}^{2}}).{{\cos }^{2}}x.2x\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer61)
  If \[y=\sin (\sqrt{\sin x+\cos x})\], then \[\frac{dy}{dx}=\] [DSSE 1987]
  A)
             \[\frac{1}{2}\frac{\cos \sqrt{\sin x+\cos x}}{\sqrt{\sin x+\cos x}}\] done clear
  B)
             \[\frac{\cos \sqrt{\sin x+\cos x}}{\sqrt{\sin x+\cos x}}\] done clear
  C)
             \[\frac{1}{2}\frac{\cos \sqrt{\sin x+\cos x}}{\sqrt{\sin x+\cos x}}.(\cos x-\sin x)\] done clear
  D)
             None of these                           done clear
  View Solution play_arrow

• question_answer62)
  If \[y=\sin \left( \frac{1+{{x}^{2}}}{1-{{x}^{2}}} \right)\], then \[\frac{dy}{dx}=\]                      [AISSE 1987]
  A)
             \[\frac{4x}{1-{{x}^{2}}}.\cos \left( \frac{1+{{x}^{2}}}{1-{{x}^{2}}} \right)\] done clear
  B)
             \[\frac{x}{{{(1-{{x}^{2}})}^{2}}}.\cos \left( \frac{1+{{x}^{2}}}{1-{{x}^{2}}} \right)\] done clear
  C)
             \[\frac{x}{(1-{{x}^{2}})}.\cos \left( \frac{1+{{x}^{2}}}{1-{{x}^{2}}} \right)\] done clear
  D)
             \[\frac{4x}{{{(1-{{x}^{2}})}^{2}}}.\cos \left( \frac{1+{{x}^{2}}}{1-{{x}^{2}}} \right)\] done clear
  View Solution play_arrow

• question_answer63)
   If \[y=\sqrt{\frac{1+\tan x}{1-\tan x}}\], then \[\frac{dy}{dx}=\] [AISSE 1981, 83, 84, 85; DSSE 1985; AI CBSE 1981, 83]
  A)
           \[\frac{1}{2}\sqrt{\frac{1-\tan x}{1+\tan x}}.{{\sec }^{2}}\left( \frac{\pi }{4}+x \right)\] done clear
  B)
             \[\sqrt{\frac{1-\tan x}{1+\tan x}}.{{\sec }^{2}}\left( \frac{\pi }{4}+x \right)\] done clear
  C)
             \[\frac{1}{2}\sqrt{\frac{1-\tan x}{1+\tan x}}.\sec \left( \frac{\pi }{4}+x \right)\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer64)
  \[\frac{d}{dx}{{({{x}^{2}}+\cos x)}^{4}}=\]                                                        [DSSE 1979]
  A)
             \[4({{x}^{2}}+\cos x)(2x-\sin x)\]                                          done clear
  B)
   \[4{{({{x}^{2}}-\cos x)}^{3}}(2x-\sin x)\]                                    done clear
  C)
             \[4{{({{x}^{2}}+\cos x)}^{3}}(2x-\sin x)\] done clear
  D)
             \[4{{({{x}^{2}}+\cos x)}^{3}}(2x+\sin x)\] done clear
  View Solution play_arrow

• question_answer65)
  \[\frac{d}{dx}\sqrt{x\sin x}=\]                              [AISSE 1985]
  A)
             \[\frac{\sin x+x\cos x}{2\sqrt{x\sin x}}\] done clear
  B)
             \[\frac{\sin x+x\cos x}{\sqrt{x\sin x}}\] done clear
  C)
             \[\frac{x\sin x+\cos x}{\sqrt{2\sin x}}\] done clear
  D)
             \[\frac{\sin x+x\cos x}{2\sqrt{x\sin x}}\] done clear
  View Solution play_arrow

• question_answer66)
  \[\frac{d}{dx}\sqrt{{{\sec }^{2}}x+\text{cose}{{\text{c}}^{2}}x}=\]                                [DSSE 1981]
  A)
             \[4\cos \text{ec 2}x.\cot 2x\] done clear
  B)
             \[-4\cos \text{ec 2}x.\cot 2x\] done clear
  C)
             \[-4\cos \text{ec }x.\cot 2x\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer67)
  \[\frac{d}{dx}\left( \frac{\sec x+\tan x}{\sec x-\tan x} \right)=\]             [DSSE 1979, 81; CBSE 1981]
  A)
             \[\frac{2\cos x}{{{(1-\sin x)}^{2}}}\] done clear
  B)
             \[\frac{\cos x}{{{(1-\sin x)}^{2}}}\] done clear
  C)
             \[\frac{2\cos x}{1-\sin x}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer68)
  \[\frac{d}{dx}\left( {{x}^{3}}{{\tan }^{2}}\frac{x}{2} \right)\]=                                          [AISSE 1979]
  A)
             \[{{x}^{3}}\tan \frac{x}{2}.{{\sec }^{2}}\frac{x}{2}+3x{{\tan }^{2}}\frac{x}{2}\]              done clear
  B)
           \[{{x}^{3}}\tan \frac{x}{2}.{{\sec }^{2}}\frac{x}{2}+3{{x}^{2}}{{\tan }^{2}}\frac{x}{2}\] done clear
  C)
             \[{{x}^{3}}{{\tan }^{2}}\frac{x}{2}.{{\sec }^{2}}\frac{x}{2}+3{{x}^{2}}{{\tan }^{2}}\frac{x}{2}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer69)
  If \[y={{\tan }^{-1}}\left( \frac{{{x}^{1/3}}+{{a}^{1/3}}}{1-{{x}^{1/3}}{{a}^{1/3}}} \right)\], then \[\frac{dy}{dx}=\]      [DSSE 1986]
  A)
           \[\frac{1}{3{{x}^{2/3}}(1+{{x}^{2/3}})}\] done clear
  B)
             \[\frac{1}{3{{x}^{2/3}}(1+{{x}^{2/3}})}\] done clear
  C)
             \[-\frac{1}{3{{x}^{2/3}}(1+{{x}^{2/3}})}\] done clear
  D)
             \[-\frac{a}{3{{x}^{2/3}}(1+{{x}^{2/3}})}\] done clear
  View Solution play_arrow

• question_answer70)
  If \[y={{\cot }^{-1}}\left( \frac{1+x}{1-x} \right)\], then \[\frac{dy}{dx}=\]                       [DSSE 1984]
  A)
             \[\frac{1}{1+{{x}^{2}}}\] done clear
  B)
           \[-\frac{1}{1+{{x}^{2}}}\] done clear
  C)
             \[\frac{2}{1+{{x}^{2}}}\] done clear
  D)
             \[-\frac{2}{1+{{x}^{2}}}\] done clear
  View Solution play_arrow

• question_answer71)
  The differential coefficient of the given function \[{{\log }_{e}}\left( \sqrt{\frac{1+\sin x}{1-\sin x}} \right)\] with respect to x is          [MP PET 1993]
  A)
             \[\cos \text{ec}\,x\] done clear
  B)
             \[\tan x\] done clear
  C)
             \[\cos x\] done clear
  D)
             \[\sec x\] done clear
  View Solution play_arrow

• question_answer72)
  \[\frac{d}{dx}\left[ \log \sqrt{\frac{1-\cos x}{1+\cos x}} \right]=\]                             [BIT Ranchi 1990]
  A)
             \[\sec x\] done clear
  B)
             \[\cos \text{ec}\,x\] done clear
  C)
             \[\cos \text{ec}\frac{x}{2}\] done clear
  D)
             \[\sec \frac{x}{2}\] done clear
  View Solution play_arrow

• question_answer73)
  \[\frac{d}{dx}\left[ {{\tan }^{-1}}\sqrt{\frac{1-\cos x}{1+\cos x}} \right]=\] [BIT Ranchi 1989; Roorkee 1989; RPET 1996]
  A)
             \[-\frac{1}{2}\] done clear
  B)
             0 done clear
  C)
             \[\frac{1}{2}\] done clear
  D)
             1 done clear
  View Solution play_arrow

• question_answer74)
  If \[f(x)={{\tan }^{-1}}\left( \frac{\sin x}{1+\cos x} \right)\],then \[f'\left( \frac{\pi }{3} \right)=\]          [BIT Ranchi 1990]
  A)
             \[\frac{1}{2(1+\cos x)}\] done clear
  B)
             \[\frac{1}{2}\] done clear
  C)
             \[\frac{1}{4}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer75)
  \[\frac{d}{dx}{{e}^{x\sin x}}=\]                          [DSSE 1979]
  A)
             \[{{e}^{x\sin x}}(x\cos x+\sin x)\] done clear
  B)
             \[{{e}^{x\sin x}}(\cos x+x\sin x)\] done clear
  C)
             \[{{e}^{x\sin x}}(\cos x+\sin x)\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer76)
  \[\frac{d}{dx}\{\log (\sec x+\tan x)\}=\]                                                         [AISSE 1982]
  A)
             \[\cos x\] done clear
  B)
             \[\sec x\] done clear
  C)
             \[\tan x\] done clear
  D)
             \[\cot x\] done clear
  View Solution play_arrow

• question_answer77)
  \[\frac{d}{dx}(x{{e}^{{{x}^{2}}}})=\]                                                                   [DSSE 1981]
  A)
             \[2{{x}^{2}}{{e}^{x}}^{2}+{{e}^{x}}^{2}\] done clear
  B)
             \[{{x}^{2}}{{e}^{x}}^{2}+{{e}^{x}}^{2}\] done clear
  C)
             \[{{e}^{x}}.2{{x}^{2}}+{{e}^{x}}^{2}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer78)
  \[\frac{d}{dx}\left[ \frac{{{e}^{ax}}}{\sin (bx+c)} \right]=\]                                   [AI CBSE 1983]
  A)
             \[\frac{{{e}^{ax}}[a\sin (bx+c)+b\cos (bx+c)]}{{{\sin }^{2}}(bx+c)}\] done clear
  B)
             \[\frac{{{e}^{ax}}[a\sin (bx+c)-b\cos (bx+c)]}{\sin (bx+c)}\] done clear
  C)
             \[\frac{{{e}^{ax}}[a\sin (bx+c)-b\cos (bx+c)]}{{{\sin }^{2}}(bx+c)}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer79)
   If \[y=\frac{{{e}^{x}}\log x}{{{x}^{2}}}\], then \[\frac{dy}{dx}=\]                                       [AI CBSE 1982]
  A)
             \[\frac{{{e}^{x}}[1+(x+2)\log x]}{{{x}^{3}}}\] done clear
  B)
             \[\frac{{{e}^{x}}[1-(x-2)\log x]}{{{x}^{4}}}\] done clear
  C)
             \[\frac{{{e}^{x}}[1-(x-2)\log x]}{{{x}^{3}}}\] done clear
  D)
             \[\frac{{{e}^{x}}[1+(x-2)\log x]}{{{x}^{3}}}\] done clear
  View Solution play_arrow

• question_answer80)
  If \[y=\frac{{{e}^{2x}}\cos x}{x\sin x},\]then \[\frac{dy}{dx}=\]                                [AI CBSE 1982]
  A)
             \[\frac{{{e}^{2x}}[(2x-1)\cot x-x\,\text{cose}{{\text{c}}^{2}}x]}{{{x}^{2}}}\] done clear
  B)
             \[\frac{{{e}^{2x}}[(2x+1)\cot x-x\,\text{cose}{{\text{c}}^{2}}x]}{{{x}^{2}}}\] done clear
  C)
             \[\frac{{{e}^{2x}}[(2x-1)\cot x+x\,\text{cose}{{\text{c}}^{2}}x]}{{{x}^{2}}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer81)
  \[\frac{d}{dx}\{{{e}^{-a{{x}^{2}}}}\log (\sin x)\}=\]                                            [AI CBSE 1984]
  A)
             \[{{e}^{-a{{x}^{2}}}}(\cot x+2ax\log \sin x)\] done clear
  B)
             \[{{e}^{-a{{x}^{2}}}}(\cot x+ax\log \sin x)\] done clear
  C)
             \[{{e}^{-a{{x}^{2}}}}(\cot x-2ax\log \sin x)\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer82)
  If \[y=\log x.{{e}^{(\tan x+{{x}^{2}})}},\]then \[\frac{dy}{dx}=\]                      [AI CBSE 1985]
  A)
             \[{{e}^{(\tan x+{{x}^{2}})}}\left[ \frac{1}{x}+({{\sec }^{2}}x+x)\log x \right]\] done clear
  B)
             \[{{e}^{(\tan x+{{x}^{2}})}}\left[ \frac{1}{x}+({{\sec }^{2}}x-x)\log x \right]\] done clear
  C)
             \[{{e}^{(\tan x+{{x}^{2}})}}\left[ \frac{1}{x}+({{\sec }^{2}}x+2x)\log x \right]\] done clear
  D)
             \[{{e}^{(\tan x+{{x}^{2}})}}\left[ \frac{1}{x}+({{\sec }^{2}}x-2x)\log x \right]\] done clear
  View Solution play_arrow

• question_answer83)
  If \[y=\sqrt{\frac{1+{{e}^{x}}}{1-{{e}^{x}}}}\], then \[\frac{dy}{dx}=\]                                         [AI CBSE 1986]
  A)
             \[\frac{{{e}^{x}}}{(1-{{e}^{x}})\sqrt{1-{{e}^{2x}}}}\] done clear
  B)
             \[\frac{{{e}^{x}}}{(1-{{e}^{x}})\sqrt{1-{{e}^{x}}}}\] done clear
  C)
             \[\frac{{{e}^{x}}}{(1-{{e}^{x}})\sqrt{1+{{e}^{2x}}}}\] done clear
  D)
             \[\frac{{{e}^{x}}}{(1-{{e}^{x}})\sqrt{1-{{e}^{2x}}}}\] done clear
  View Solution play_arrow

• question_answer84)
  \[\frac{d}{dx}\left\{ {{e}^{x}}\log (1+{{x}^{2}}) \right\}=\]                                [AI CBSE 1987]
  A)
             \[{{e}^{x}}\left[ \log (1+{{x}^{2}})+\frac{2x}{1+{{x}^{2}}} \right]\] done clear
  B)
             \[{{e}^{x}}\left[ \log (1+{{x}^{2}})-\frac{2x}{1+{{x}^{2}}} \right]\] done clear
  C)
             \[{{e}^{x}}\left[ \log (1+{{x}^{2}})+\frac{x}{1+{{x}^{2}}} \right]\]      done clear
  D)
             \[{{e}^{x}}\left[ \log (1+{{x}^{2}})-\frac{x}{1+{{x}^{2}}} \right]\] done clear
  View Solution play_arrow

• question_answer85)
  If \[y=\frac{{{e}^{2x}}+{{e}^{-2x}}}{{{e}^{2x}}-{{e}^{-2x}}}\], then \[\frac{dy}{dx}=\]                   [AI CBSE 1988]
  A)
             \[\frac{-8}{{{({{e}^{2x}}-{{e}^{-2x}})}^{2}}}\] done clear
  B)
             \[\frac{8}{{{({{e}^{2x}}-{{e}^{-2x}})}^{2}}}\] done clear
  C)
             \[\frac{-4}{{{({{e}^{2x}}-{{e}^{-2x}})}^{2}}}\] done clear
  D)
             \[\frac{4}{{{({{e}^{2x}}-{{e}^{-2x}})}^{2}}}\] done clear
  View Solution play_arrow

• question_answer86)
  If \[y=\frac{2{{(x-\sin x)}^{3/2}}}{\sqrt{x}}\], then \[\frac{dy}{dx}=\]                 [Roorkee 1971]
  A)
             \[\frac{2{{(x-\sin x)}^{3/2}}}{\sqrt{x}}\left[ \frac{3}{2}.\frac{1-\cos x}{1-\sin x}-\frac{1}{2x} \right]\] done clear
  B)
             \[\frac{2{{(x-\sin x)}^{3/2}}}{\sqrt{x}}\left[ \frac{3}{2}.\frac{1-\cos x}{x-\sin x}-\frac{1}{2x} \right]\] done clear
  C)
             \[\frac{2{{(x-\sin x)}^{1/2}}}{\sqrt{x}}\left[ \frac{3}{2}.\frac{1-\cos x}{x-\sin x}-\frac{1}{2x} \right]\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer87)
  \[\frac{d}{dx}\left( {{\cos }^{-1}}\sqrt{\frac{1+\cos x}{2}} \right)=\]                                [AI CBSE 1982]
  A)
             1 done clear
  B)
             \[\frac{1}{2}\] done clear
  C)
             \[\frac{1}{3}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer88)
  If \[y={{\tan }^{-1}}\left( \frac{\sqrt{a}-\sqrt{x}}{1+\sqrt{ax}} \right)\], then \[\frac{dy}{dx}=\]          [AI CBSE 1988]
  A)
             \[\frac{1}{2(1+x)\sqrt{x}}\] done clear
  B)
             \[\frac{1}{(1+x)\sqrt{x}}\] done clear
  C)
             \[-\frac{1}{2(1+x)\sqrt{x}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer89)
  If \[y={{\sec }^{-1}}\left( \frac{x+1}{x-1} \right)+{{\sin }^{-1}}\left( \frac{x-1}{x+1} \right)\], then \[\frac{dy}{dx}=\]          [MNR 1984]
  A)
             0 done clear
  B)
             1 done clear
  C)
             2 done clear
  D)
             3 done clear
  View Solution play_arrow

• question_answer90)
  \[\frac{d}{dx}({{\log }_{e}}x)({{\log }_{a}}x)]=\]                          
  A)
             \[\frac{{{\log }_{a}}x}{x}\] done clear
  B)
   \[\frac{{{\log }_{x}}x}{x}\] done clear
  C)
             \[\frac{2\log x}{x}\] done clear
  D)
             \[\frac{2{{\log }_{a}}x}{x}\] done clear
  View Solution play_arrow

• question_answer91)
  \[\frac{d}{dx}\left\{ \log \left( \frac{{{e}^{x}}}{1+{{e}^{x}}} \right) \right\}=\]   
  A)
             \[\frac{1}{1-{{e}^{x}}}\] done clear
  B)
             \[-\frac{1}{1+{{e}^{x}}}\] done clear
  C)
             \[-\frac{1}{1-{{e}^{x}}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer92)
   \[\frac{d}{dx}\left[ \frac{2}{\pi }\sin {{x}^{0}} \right]=\]
  A)
             \[\frac{\pi }{180}\cos {{x}^{0}}\] done clear
  B)
             \[\frac{1}{90}\cos {{x}^{0}}\] done clear
  C)
             \[\frac{\pi }{90}\cos {{x}^{0}}\] done clear
  D)
             \[\frac{2}{90}\cos {{x}^{0}}\] done clear
  View Solution play_arrow

• question_answer93)
  \[\frac{d}{dx}\left[ \log \sqrt{\sin \sqrt{{{e}^{x}}}} \right]\]=       
  A)
             \[\frac{1}{4}{{e}^{x/2}}\cot ({{e}^{x/2}})\] done clear
  B)
             \[{{e}^{x/2}}\cot ({{e}^{x/2}})\] done clear
  C)
             \[\frac{1}{4}{{e}^{x}}\cot \,({{e}^{x}})\] done clear
  D)
             \[\frac{1}{2}{{e}^{x/2}}\cot \,({{e}^{x/2}})\] done clear
  View Solution play_arrow

• question_answer94)
  If \[f(x)=\,|x|,\]then \[f'(0)=\]                                     [MNR 1982]
  A)
             0 done clear
  B)
             1 done clear
  C)
             x done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer95)
  At \[x=\sqrt{\frac{\pi }{2}},\frac{d}{dx}\cos (\sin {{x}^{2}})\]=
  A)
             ?1 done clear
  B)
             1 done clear
  C)
             0 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer96)
   \[\frac{d}{dx}[{{\tan }^{-1}}(\cot x)+{{\cot }^{-1}}(\tan x)]=\]
  A)
             0 done clear
  B)
             1 done clear
  C)
             ? 1 done clear
  D)
             ? 2 done clear
  View Solution play_arrow

• question_answer97)
  \[\frac{d}{dx}[{{e}^{ax}}\cos (bx+c)]\]=                                                               [AISSE 1989]
  A)
             \[{{e}^{ax}}[a\cos (bx+c)-b\sin (bx+c)]\] done clear
  B)
             \[{{e}^{ax}}[a\sin (bx+c)-b\cos (bx+c)]\] done clear
  C)
             \[{{e}^{ax}}[\cos (bx+c)-\sin (bx+c)]\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer98)
  If \[y=\log \log x\], then \[{{e}^{y}}\frac{dy}{dx}=\]                                    [MP PET 1994, 95]
  A)
             \[\frac{1}{x\log x}\] done clear
  B)
             \[\frac{1}{x}\] done clear
  C)
             \[\frac{1}{\log x}\] done clear
  D)
             \[{{e}^{y}}\] done clear
  View Solution play_arrow

• question_answer99)
  If \[y={{\sin }^{-1}}\left( \frac{19}{20}x \right)+{{\cos }^{-1}}\left( \frac{19}{20}x \right)\], then \[\frac{dy}{dx}=\]
  A)
             0 done clear
  B)
             1 done clear
  C)
             ? 1 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer100)
  If \[y=(1+{{x}^{1/4}})(1+{{x}^{1/2}})(1-{{x}^{1/4}})\], then \[\frac{dy}{dx}\]= [MP PET 1994]
  A)
             1 done clear
  B)
             ? 1 done clear
  C)
             x done clear
  D)
             \[\sqrt{x}\] done clear
  View Solution play_arrow

• question_answer101)
  If \[y=\frac{{{a}^{{{\cos }^{-1}}x}}}{1+{{a}^{{{\cos }^{-1}}x}}}\]and \[z={{a}^{{{\cos }^{-1}}x}}\],  then \[\frac{dy}{dx}\]= [MP PET 1994]
  A)
             \[\frac{1}{1+{{a}^{{{\cos }^{-1}}x}}}\] done clear
  B)
             \[-\frac{1}{1+{{a}^{{{\cos }^{-1}}x}}}\] done clear
  C)
             \[\frac{1}{{{(1+{{a}^{{{\cos }^{-1}}x}})}^{2}}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer102)
  If \[f(x)=(x-{{x}_{0}})g(x)\], where \[g(x)\] is continuous at \[{{x}_{0}}\], then \[f'({{x}_{0}})\] is equal to           
  A)
             0 done clear
  B)
             \[{{x}_{0}}\] done clear
  C)
             \[g({{x}_{0}})\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer103)
  If\[y={{\log }_{\sin x}}(\tan x),\] then \[{{\left( \frac{dy}{dx} \right)}_{\pi /4}}=\]
  A)
             \[\frac{4}{\log 2}\] done clear
  B)
             \[-4\log 2\] done clear
  C)
             \[\frac{-4}{\log 2}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer104)
  If \[y={{\log }_{2}}[{{\log }_{2}}(x)]\], then \[\frac{dy}{dx}\]is equal to
  A)
             \[\frac{{{\log }_{2}}e}{x{{\log }_{e}}x}\] done clear
  B)
             \[\frac{1}{{{\log }_{e}}x{{\log }_{e}}2}\] done clear
  C)
             \[\frac{1}{{{\log }_{e}}{{(2x)}^{x}}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer105)
   \[\frac{d}{dx}({{e}^{{{x}^{3}}}})\] is equal to                                                         [RPET 1995]
  A)
             \[3x{{e}^{{{x}^{3}}}}\] done clear
  B)
    \[3{{x}^{2}}{{e}^{{{x}^{3}}}}\] done clear
  C)
             \[3x{{\left( {{e}^{{{x}^{3}}}} \right)}^{2}}\] done clear
  D)
   \[2{{x}^{2}}{{e}^{{{x}^{3}}}}\] done clear
  View Solution play_arrow

• question_answer106)
  \[\frac{d}{dx}({{\sin }^{-1}}x)\] is equal to                                                                    [RPET 1995]
  A)
           \[\frac{1}{\sqrt{1-{{x}^{2}}}}\] done clear
  B)
    \[-\frac{1}{\sqrt{1-{{x}^{2}}}}\] done clear
  C)
             \[\frac{1}{\sqrt{1+{{x}^{2}}}}\] done clear
  D)
    \[\frac{-1}{\sqrt{1+{{x}^{2}}}}\] done clear
  View Solution play_arrow

• question_answer107)
  If \[y={{\tan }^{-1}}\sqrt{\frac{1+\cos x}{1-\cos x}}\], then \[\frac{dy}{dx}\] is equal to [Roorkee 1995]
  A)
             0 done clear
  B)
             \[-\frac{1}{2}\] done clear
  C)
             ½ done clear
  D)
             1 done clear
  View Solution play_arrow

• question_answer108)
  If \[y=\frac{{{\sin }^{-1}}x}{\sqrt{1-{{x}^{2}}}}\], then \[(1-{{x}^{2}})\frac{dy}{dx}\] is equal to [RPET 1995]
  A)
             \[x+y\] done clear
  B)
             \[1+xy\] done clear
  C)
             1? xy done clear
  D)
             \[xy-2\] done clear
  View Solution play_arrow

• question_answer109)
  Differential coefficient of \[{{\sec }^{-1}}x\] is            [RPET 1995]
  A)
   \[\frac{1}{x\sqrt{1-{{x}^{2}}}}\] done clear
  B)
   \[-\frac{1}{x\sqrt{1-{{x}^{2}}}}\] done clear
  C)
   \[\frac{1}{x\sqrt{{{x}^{2}}-1}}\] done clear
  D)
   \[\frac{-1}{x\sqrt{{{x}^{2}}-1}}\] done clear
  View Solution play_arrow

• question_answer110)
  If \[f(2)=4\], \[f'(2)=1\]then \[\underset{x\to 2}{\mathop{\lim }}\,\frac{xf(2)-2f(x)}{x-2}=\] [RPET 1995, 2000]
  A)
             1 done clear
  B)
             2 done clear
  C)
             3 done clear
  D)
              ?2 done clear
  View Solution play_arrow

• question_answer111)
  \[\frac{d}{dx}\left[ \log \left( x+\frac{1}{x} \right) \right]=\]                                            [MP PET 1995]
  A)
             \[\left( x+\frac{1}{x} \right)\] done clear
  B)
             \[\frac{\left( 1+\frac{1}{{{x}^{2}}} \right)}{\left( 1+\frac{1}{x} \right)}\] done clear
  C)
             \[\frac{\left( 1-\frac{1}{{{x}^{2}}} \right)}{\left( x+\frac{1}{x} \right)}\] done clear
  D)
             \[\left( 1+\frac{1}{x} \right)\] done clear
  View Solution play_arrow

• question_answer112)
    If \[y={{\sin }^{-1}}\sqrt{x}\],  then \[\frac{dy}{dx}=\]                              [MP PET 1995]
  A)
             \[\frac{2}{\sqrt{x}\sqrt{1-x}}\] done clear
  B)
             \[\frac{-2}{\sqrt{x}\sqrt{1-x}}\] done clear
  C)
             \[\frac{1}{2\sqrt{x}\sqrt{1-x}}\] done clear
  D)
             \[\frac{1}{\sqrt{1-x}}\] done clear
  View Solution play_arrow

• question_answer113)
  If \[y={{\sin }^{-1}}\sqrt{(1-x)}+{{\cos }^{-1}}\sqrt{x}\], then \[\frac{dy}{dx}=\]
  A)
             \[\frac{1}{\sqrt{x(1-x)}}\] done clear
  B)
             \[\frac{-1}{\sqrt{x(1-x)}}\] done clear
  C)
             \[\frac{1}{\sqrt{x(1+x)}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer114)
  If \[y={{x}^{n}}\log x+x{{(\log x)}^{n}}\], then \[\frac{dy}{dx}=\]
  A)
             \[{{x}^{n-1}}(1+n\log x)+{{(\log x)}^{n-1}}[n+\log x]\] done clear
  B)
             \[{{x}^{n-2}}(1+n\log x)+{{(\log x)}^{n-1}}[n+\log x]\] done clear
  C)
             \[{{x}^{n-1}}(1+n\log x)+{{(\log x)}^{n-1}}[n-\log x]\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer115)
  If \[y\sqrt{{{x}^{2}}+1}=\log \left\{ \sqrt{{{x}^{2}}+1}-x \right\}\], then\[({{x}^{2}}+1)\frac{dy}{dx}+xy+1=\]  [Roorkee 1978; Kurukshetra CEE 1998]
  A)
             0 done clear
  B)
             1 done clear
  C)
             2 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer116)
  The derivative of tanx ? x with respect to x is  [SCRA 1996]
  A)
             \[1-{{\tan }^{2}}x\] done clear
  B)
             tan x done clear
  C)
             \[-{{\tan }^{2}}x\] done clear
  D)
             \[{{\tan }^{2}}x\] done clear
  View Solution play_arrow

• question_answer117)
  If \[f(x)=({{\log }_{\cot x}}\tan x){{({{\log }_{\tan x}}\cot x)}^{-1}},\]then \[f'(2)=\]
  A)
             2 done clear
  B)
             0 done clear
  C)
             \[\frac{1}{2}\] done clear
  D)
             ? 2 done clear
  View Solution play_arrow

• question_answer118)
  If \[f(x)=3{{e}^{{{x}^{2}}}}\],then \[f'(x)-2xf(x)+\frac{1}{3}f(0)-f'(0)=\]
  A)
             0 done clear
  B)
             1 done clear
  C)
             \[\frac{7}{3}{{e}^{{{x}^{2}}}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer119)
  If \[y={{\log }_{\cos x}}\sin x\], then \[\frac{dy}{dx}\]is equal to
  A)
             \[\frac{\cot x\log \cos x+\tan x\log \sin x}{{{(\log \cos x)}^{2}}}\] done clear
  B)
             \[\frac{\tan x\log \cos x+\cot x\log \sin x}{{{(\log \cos x)}^{2}}}\] done clear
  C)
             \[\frac{\cot x\log \cos x+\tan x\log \sin x}{{{(\log \sin x)}^{2}}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer120)
  \[\frac{d}{dx}\left[ \log \left\{ {{e}^{x}}{{\left( \frac{x+2}{x-2} \right)}^{3/4}} \right\} \right]\] equals                              
  A)
             \[\frac{{{x}^{2}}-7}{{{x}^{2}}-4}\] done clear
  B)
             1 done clear
  C)
             \[\frac{{{x}^{2}}+1}{{{x}^{2}}-4}\] done clear
  D)
             \[{{e}^{x}}\frac{{{x}^{2}}-1}{{{x}^{2}}-4}\] done clear
  View Solution play_arrow

• question_answer121)
  For the curve \[\sqrt{x}+\sqrt{y}=1,\frac{dy}{dx}\]at \[\left( \frac{1}{4},\frac{1}{4} \right)\]is [Karnataka CET 1993]
  A)
             ½ done clear
  B)
             1 done clear
  C)
             ?1 done clear
  D)
             2 done clear
  View Solution play_arrow

• question_answer122)
  Differential coefficient of \[\sqrt{\sec \sqrt{x}}\]is     [MP PET 1996]
  A)
   \[\frac{1}{4\sqrt{x}}{{(\sec \sqrt{x})}^{3/2}}\sin \sqrt{x}\] done clear
  B)
   \[\frac{1}{4\sqrt{x}}\sec \sqrt{x}\sin \sqrt{x}\]  done clear
  C)
   \[\frac{1}{2}\sqrt{x}{{(\sec \sqrt{x})}^{3/2}}\sin \sqrt{x}\] done clear
  D)
    \[\frac{1}{2}\sqrt{x}\sec \sqrt{x}\sin \sqrt{x}\] done clear
  View Solution play_arrow

• question_answer123)
  If \[y={{e}^{(1+{{\log }_{e}}x)}}\], then the value of \[\frac{dy}{dx}=\] [MP PET 1996; Pb. CET 2001]
  A)
             e done clear
  B)
             1 done clear
  C)
             0 done clear
  D)
             \[{{\log }_{e}}x\,\,{{e}^{{{\log }_{e}}ex}}\] done clear
  View Solution play_arrow

• question_answer124)
  For the function \[f(x)={{x}^{2}}-6x+8,2\le x\le 4\], the value of x for which \[f'(x)\] vanishes, is                                          [MP PET 1996]
  A)
             \[\frac{9}{4}\] done clear
  B)
             \[\frac{5}{2}\] done clear
  C)
             3 done clear
  D)
             \[\frac{7}{2}\] done clear
  View Solution play_arrow

• question_answer125)
  If \[f(x)={{e}^{x}}g(x),g(0)=2,g'(0)=1\], then \[f'(0)\]is
  A)
             1 done clear
  B)
             3 done clear
  C)
             2 done clear
  D)
             0 done clear
  View Solution play_arrow

• question_answer126)
  If \[y={{e}^{x}}\log x\], then \[\frac{dy}{dx}\]is                                                            [SCRA 1996]
  A)
   \[\frac{{{e}^{x}}}{x}\] done clear
  B)
   \[{{e}^{x}}\left( \frac{1}{x}+x\log x \right)\] done clear
  C)
   \[{{e}^{x}}\left( \frac{1}{x}+\log x \right)\] done clear
  D)
   \[\frac{{{e}^{x}}}{\log x}\] done clear
  View Solution play_arrow

• question_answer127)
  If \[y={{\cot }^{-1}}\left[ \frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}} \right]\], then \[\frac{dy}{dx}=\]
  A)
   \[\frac{1}{2}\] done clear
  B)
             \[\frac{2}{3}\] done clear
  C)
             3 done clear
  D)
             1 done clear
  View Solution play_arrow

• question_answer128)
  If \[y=\sec {{x}^{0}}\], then \[\frac{dy}{dx}=\]                                                             [MP PET 1997]
  A)
             \[\sec x\tan x\] done clear
  B)
             \[\sec {{x}^{o}}\tan {{x}^{o}}\] done clear
  C)
   \[\frac{\pi }{180}\sec {{x}^{o}}\tan {{x}^{o}}\] done clear
  D)
    \[\frac{180}{\pi }\sec {{x}^{o}}\tan {{x}^{o}}\] done clear
  View Solution play_arrow

• question_answer129)
  If \[y=\sqrt{\sin \sqrt{x}}\],  then \[\frac{dy}{dx}=\]                                                        [MP PET 1997]
  A)
             \[\frac{1}{2\sqrt{\cos \sqrt{x}}}\] done clear
  B)
   \[\frac{\sqrt{\cos \sqrt{x}}}{2x}\] done clear
  C)
   \[\frac{\cos \sqrt{x}}{4\sqrt{x}\sqrt{\sin \sqrt{x}}}\] done clear
  D)
   \[\frac{1}{2\sqrt{\sin x}}\] done clear
  View Solution play_arrow

• question_answer130)
  If\[y={{\log }_{10}}{{x}^{2}}\], then \[\frac{dy}{dx}\]  is equal to
  A)
   \[\frac{2}{x}\] done clear
  B)
    \[\frac{2}{x{{\log }_{e}}10}\] done clear
  C)
   \[\frac{1}{x{{\log }_{e}}10}\] done clear
  D)
    \[\frac{1}{10x}\] done clear
  View Solution play_arrow

• question_answer131)
  If \[y={{3}^{{{x}^{2}}}}\], then \[\frac{dy}{dx}\] is equal to
  A)
             \[({{x}^{2}}){{3}^{{{x}^{2}}-1}}\] done clear
  B)
             \[3{{x}^{2}}.2x\] done clear
  C)
             \[{{3}^{{{x}^{2}}}}.2x.\log 3\] done clear
  D)
             \[({{x}^{2}}-1).3\] done clear
  View Solution play_arrow

• question_answer132)
  The first derivative of the function \[(\sin 2x\cos 2x\cos 3x+{{\log }_{2}}{{2}^{x+3}})\] with respect to x at \[x=\pi \]is                              [MP PET 1998]
  A)
             2 done clear
  B)
             ?1 done clear
  C)
             \[-2+{{2}^{\pi }}{{\log }_{e}}2\] done clear
  D)
             \[-2+{{\log }_{e}}2\] done clear
  View Solution play_arrow

• question_answer133)
  The values of x, at which the first derivative of the function \[{{\left( \sqrt{x}+\frac{1}{\sqrt{x}} \right)}^{2}}\]w.r.t. x is \[\frac{3}{4}\], are [MP PET 1998]
  A)
                     \[\pm \,2\] done clear
  B)
             \[\pm \frac{1}{2}\] done clear
  C)
                     \[\pm \frac{\sqrt{3}}{2}\] done clear
  D)
   \[\pm \frac{2}{\sqrt{3}}\] done clear
  View Solution play_arrow

• question_answer134)
   If \[y=\frac{{{(1-x)}^{2}}}{{{x}^{2}}}\], then \[\frac{dy}{dx}\]is                                          [MP PET 1999]
  A)
   \[\frac{2}{{{x}^{2}}}+\frac{2}{{{x}^{3}}}\] done clear
  B)
   \[-\frac{2}{{{x}^{2}}}+\frac{2}{{{x}^{3}}}\] done clear
  C)
   \[-\frac{2}{{{x}^{2}}}-\frac{2}{{{x}^{3}}}\] done clear
  D)
    \[-\frac{2}{{{x}^{3}}}+\frac{2}{{{x}^{2}}}\] done clear
  View Solution play_arrow

• question_answer135)
  If \[pv=81\], then \[\frac{dp}{dv}\] is at v = 9 equal to [MP PET 1999]
  A)
             1 done clear
  B)
           ?1 done clear
  C)
             2 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer136)
  \[\frac{d}{dx}(\sin 2{{x}^{2}})\] equals                                                                         [RPET 1996]
  A)
             \[4x\cos \,(2{{x}^{2}})\] done clear
  B)
             \[2\sin {{x}^{2}}\cos {{x}^{2}}\] done clear
  C)
             \[4x\sin ({{x}^{2}})\] done clear
  D)
             \[4x\sin ({{x}^{2}})\cos ({{x}^{2}})\] done clear
  View Solution play_arrow

• question_answer137)
   \[\frac{d}{dx}\cos \,{{\text{h}}^{-1}}(\sec x)=\]                                                           [RPET 1997]
  A)
             sec x done clear
  B)
             sin x done clear
  C)
             tan x done clear
  D)
             cosec x done clear
  View Solution play_arrow

• question_answer138)
  If \[f(x)=\frac{1}{\sqrt{{{x}^{2}}+{{a}^{2}}}+\sqrt{{{x}^{2}}+{{b}^{2}}}}\], then \[{f}'(x)\] is equal to [Kurukshetra  CEE 1998]
  A)
           \[\frac{x}{({{a}^{2}}-{{b}^{2}})}\left[ \frac{1}{\sqrt{{{x}^{2}}+{{a}^{2}}}}-\frac{1}{\sqrt{{{x}^{2}}+{{b}^{2}}}} \right]\] done clear
  B)
             \[\frac{x}{({{a}^{2}}+{{b}^{2}})}\left[ \frac{1}{\sqrt{{{x}^{2}}+{{a}^{2}}}}-\frac{2}{\sqrt{{{x}^{2}}+{{b}^{2}}}} \right]\] done clear
  C)
             \[\frac{x}{({{a}^{2}}-{{b}^{2}})}\left[ \frac{1}{\sqrt{{{x}^{2}}+{{a}^{2}}}}+\frac{1}{\sqrt{{{x}^{2}}+{{b}^{2}}}} \right]\] done clear
  D)
             \[({{a}^{2}}+{{b}^{2}})\left[ \frac{1}{\sqrt{{{x}^{2}}+{{a}^{2}}}}-\frac{2}{\sqrt{{{x}^{2}}+{{b}^{2}}}} \right]\] done clear
  View Solution play_arrow

• question_answer139)
  The derivative of \[f(x)=x|x|\] is               [SCRA 1996]
  A)
             \[2x\] done clear
  B)
             ? 2x done clear
  C)
             \[2{{x}^{2}}\] done clear
  D)
             \[2|x|\] done clear
  View Solution play_arrow

• question_answer140)
  The derivative of \[y=1-|x|\]at \[x=0\]is [SCRA 1996]
  A)
             0 done clear
  B)
             1 done clear
  C)
             ?1 done clear
  D)
           Does not exist done clear
  View Solution play_arrow

• question_answer141)
  The derivative of \[\sqrt{\sqrt{x}+1}\]is                                                            [SCRA 1996]
  A)
              \[\frac{1}{\sqrt{x}(\sqrt{x}+1)}\] done clear
  B)
   \[\frac{1}{\sqrt{x}\sqrt{x+1}}\] done clear
  C)
             \[\frac{4}{\sqrt{x(\sqrt{x}+1)}}\] done clear
  D)
   \[\frac{1}{4\sqrt{x(\sqrt{x}+1)}}\] done clear
  View Solution play_arrow

• question_answer142)
  If \[y={{e}^{\sqrt{x}}}\], then  \[\frac{dy}{dx}\]equals                                                [SCRA 1996]
  A)
             \[\frac{{{e}^{\sqrt{x}}}}{2\sqrt{x}}\] done clear
  B)
    \[\frac{\sqrt{x}}{{{e}^{\sqrt{x}}}}\] done clear
  C)
             \[\frac{x}{{{e}^{\sqrt{x}}}}\] done clear
  D)
   \[\frac{2\sqrt{x}}{{{e}^{\sqrt{x}}}}\] done clear
  View Solution play_arrow

• question_answer143)
  If \[f(x)={{\cos }^{-1}}\left[ \frac{1-{{(\log x)}^{2}}}{1+{{(\log x)}^{2}}} \right]\,,\]then the value of \[f'(e)=\]            [Karnataka CET 1999; Pb. CET 2000]
  A)
             1 done clear
  B)
             1/e done clear
  C)
             2/e done clear
  D)
             \[\frac{2}{{{e}^{2}}}\] done clear
  View Solution play_arrow

• question_answer144)
  The derivative of \[f(x)=\,|{{x}^{2}}-x|\] at x = 2 is [AMU 1999]
  A)
             ? 3 done clear
  B)
             0 done clear
  C)
           3 done clear
  D)
             Not defined done clear
  View Solution play_arrow

• question_answer145)
  If \[f(1)=3,\,{f}'(1)=2,\]then \[\frac{d}{dx}\{\log f\,({{e}^{x}}+2x)\}\] at \[x=0\] is                                                               [AMU 1999]
  A)
             2 / 3 done clear
  B)
             3 / 2 done clear
  C)
           2 done clear
  D)
             0 done clear
  View Solution play_arrow

• question_answer146)
  \[\frac{d}{dx}{{\log }_{\sqrt{x}}}(1/x)\]is equal to                                                         [AMU 1999]
  A)
             \[-\frac{1}{2\sqrt{x}}\] done clear
  B)
             ? 2 done clear
  C)
             \[-\frac{1}{{{x}^{2}}\sqrt{x}}\] done clear
  D)
             0 done clear
  View Solution play_arrow

• question_answer147)
  The value of \[\frac{d}{dx}[|x-1|+|x-5|]\] at \[x=3\] is [MP PET 2000]
  A)
             ? 2 done clear
  B)
             0 done clear
  C)
             2 done clear
  D)
             4 done clear
  View Solution play_arrow

• question_answer148)
  \[\frac{d}{dx}\left[ \left( \frac{{{\tan }^{2}}2x-{{\tan }^{2}}x}{1-{{\tan }^{2}}2x{{\tan }^{2}}x} \right)\cot 3x \right]\]                 [AMU 2000]
  A)
             \[\tan 2x\,\tan x\] done clear
  B)
             \[\tan 3x\tan x\] done clear
  C)
             \[{{\sec }^{2}}x\] done clear
  D)
             \[\sec x\tan x\] done clear
  View Solution play_arrow

• question_answer149)
  If \[y={{\tan }^{-1}}\left( \frac{\sqrt{x}-x}{1+{{x}^{3/2}}} \right),\]then \[y'(1)\] is              [AMU 2000]
  A)
             0 done clear
  B)
             \[\frac{1}{2}\] done clear
  C)
             ? 1 done clear
  D)
             \[-\frac{1}{4}\] done clear
  View Solution play_arrow

• question_answer150)
  \[{{10}^{-x\,\tan x}}\left[ \frac{d}{dx}({{10}^{x\tan x}}) \right]\] is equal to                          [AMU 2000]
  A)
             \[\tan x\,+x\,\,{{\sec }^{2}}x\] done clear
  B)
             \[\ln \,10\,(\tan x+x{{\sec }^{2}}x)\] done clear
  C)
             \[\ln \,10\,\left( \tan x+\frac{x}{{{\cos }^{2}}x}+\tan x\sec x \right)\] done clear
  D)
             \[x\tan x\,\text{ln}\,\,10\] done clear
  View Solution play_arrow

• question_answer151)
  If \[f(x)\] has a derivative at \[x=a,\]then \[\underset{x\to a}{\mathop{\lim }}\,\frac{xf(a)-af(x)}{x-a}\] is equal to                                   [AMU 2000]
  A)
             \[f(a)-a\,f\,'(a)\] done clear
  B)
             \[a\,f(a)-f\,'(a)\] done clear
  C)
             \[f(a)+f'(a)\] done clear
  D)
             \[a\,f(a)+f\,'(a)\] done clear
  View Solution play_arrow

• question_answer152)
  Given that \[\frac{d}{dx}f(x)=f\,'(x)\]. The relationship \[f\,'(a+b)=f\,'(a)+f\,'(b)\] is valid if \[f(x)\] is equal to [AMU 2000]
  A)
             \[x\] done clear
  B)
             \[{{x}^{2}}\] done clear
  C)
             \[{{x}^{3}}\] done clear
  D)
             \[{{x}^{4}}\] done clear
  View Solution play_arrow

• question_answer153)
  The derivative of \[f(x)=|x{{|}^{3}}\] at \[x=0\] is [RPET 2001; Kurukshetra CEE 2002]
  A)
             0 done clear
  B)
             1 done clear
  C)
             ?1 done clear
  D)
             Not defined done clear
  View Solution play_arrow

• question_answer154)
  If \[y=\sqrt{\sin x+y},\] then \[\frac{dy}{dx}\] equals to [RPET 2001]
  A)
             \[\frac{\sin x}{2y-1}\] done clear
  B)
             \[\frac{\cos x}{2y-1}\] done clear
  C)
             \[\frac{\sin x}{2y+1}\] done clear
  D)
             \[\frac{\cos x}{2y+1}\] done clear
  View Solution play_arrow

• question_answer155)
  If \[y=(1+{{x}^{2}}){{\tan }^{-1}}x-x,\]then \[\frac{dy}{dx}=\]             [Karnataka CET 2001]
  A)
             \[{{\tan }^{-1}}x\] done clear
  B)
             \[2x{{\tan }^{-1}}x\] done clear
  C)
             \[2x{{\tan }^{-1}}x-1\] done clear
  D)
             \[\frac{2x}{{{\tan }^{-1}}x}\] done clear
  View Solution play_arrow

• question_answer156)
  If \[x=y\sqrt{1-{{y}^{2}},}\]then \[\frac{dy}{dx}=\]                                                    [MP PET 2001]
  A)
             0 done clear
  B)
             x done clear
  C)
             \[\frac{\sqrt{1-{{y}^{2}}}}{1-2{{y}^{2}}}\] done clear
  D)
             \[\frac{\sqrt{1-{{y}^{2}}}}{1+2{{y}^{2}}}\] done clear
  View Solution play_arrow

• question_answer157)
  If \[y={{\tan }^{-1}}\left[ \frac{\sin x+\cos x}{\cos x-\sin x} \right]\,,\] then \[\frac{dy}{dx}\] is               [UPSEAT 2001]
  A)
             \[1/2\] done clear
  B)
             \[\pi /4\] done clear
  C)
             0 done clear
  D)
             1 done clear
  View Solution play_arrow

• question_answer158)
  If \[y=\frac{a+b{{x}^{3/2}}}{{{x}^{5/4}}}\] and \[{y}'=0\] at \[x=5\], then the ratio \[a:b\] is equal to                                            [AMU 2001]
  A)
             \[\sqrt{5}:1\] done clear
  B)
             5 : 2 done clear
  C)
             3 : 5                                           done clear
  D)
             1 : 2 done clear
  View Solution play_arrow

• question_answer159)
  \[\frac{d}{dx}\left[ {{\tan }^{-1}}\left( \frac{a-x}{1+ax} \right) \right]=\] [Karnataka CET 2001; Pb. CET 2001]
  A)
             \[-\frac{1}{1+{{x}^{2}}}\] done clear
  B)
             \[\frac{1}{1+{{a}^{2}}}-\frac{1}{1+{{x}^{2}}}\] done clear
  C)
             \[\frac{1}{1+{{\left( \frac{a-x}{1+ax} \right)}^{2}}}\] done clear
  D)
             \[\frac{-1}{\sqrt{1-{{\left( \frac{a-x}{1+ax} \right)}^{2}}}}\] done clear
  View Solution play_arrow

• question_answer160)
  \[\frac{d}{dx}\left[ \log \left\{ {{e}^{x}}{{\left( \frac{x-2}{x+2} \right)}^{3/4}} \right\} \right]\] equals to          [RPET 2001]
  A)
             1 done clear
  B)
             \[\frac{{{x}^{2}}+1}{{{x}^{2}}-4}\] done clear
  C)
             \[\frac{{{x}^{2}}-1}{{{x}^{2}}-4}\] done clear
  D)
             \[{{e}^{x}}\frac{{{x}^{2}}-1}{{{x}^{2}}-4}\] done clear
  View Solution play_arrow

• question_answer161)
  If \[y=\sec ({{\tan }^{-1}}x),\]then \[\frac{dy}{dx}\] is [DCE 2002; Kurukshetra CEE 2001]
  A)
             \[\frac{x}{\sqrt{1+{{x}^{2}}}}\] done clear
  B)
             \[\frac{-x}{\sqrt{1+{{x}^{2}}}}\] done clear
  C)
             \[\frac{x}{\sqrt{1-{{x}^{2}}}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer162)
  The differential coefficient of the function \[|x-1|+|x-3|\] at the point \[x=2\] is           [RPET 2002; Pb. CET 2000, 04]
  A)
             ? 2 done clear
  B)
             0 done clear
  C)
             2 done clear
  D)
             Undefined done clear
  View Solution play_arrow

• question_answer163)
  If \[f(x)\] is a differentiable function, then \[\underset{x\to a}{\mathop{\lim }}\,\frac{af(x)-xf(a)}{x-a}\] is                    [UPSEAT 2002]
  A)
             \[a{f}'\,(a)-f\,(a)\] done clear
  B)
             \[af\,(a)-f'(a)\] done clear
  C)
             \[a{f}'\,(a)+f\,(a)\] done clear
  D)
             \[af\,(a)+f'(a)\] done clear
  View Solution play_arrow

• question_answer164)
  If \[x=\exp \left\{ {{\tan }^{-1}}\left( \frac{y-{{x}^{2}}}{{{x}^{2}}} \right) \right\}\,\,\], then \[\frac{dy}{dx}\]equals [MP PET 2002]
  A)
             \[2x\,[1+\tan \,(\log x)]+x{{\sec }^{2}}(\log x)\] done clear
  B)
             \[x\,[1+\tan \,(\log x)]+{{\sec }^{2}}(\log x)\] done clear
  C)
             \[2x\,[1+\tan \,(\log x)]+{{x}^{2}}\,\,{{\sec }^{2}}(\log x)\] done clear
  D)
             \[2x\,[1+\tan \,(\log x)]+{{\sec }^{2}}(\log x)\] done clear
  View Solution play_arrow

• question_answer165)
  If \[f(x)=\sqrt{ax}+\frac{{{a}^{2}}}{\sqrt{ax}},\]then \[f'(a)=\]                  [EAMCET 2002]
  A)
             ? 1 done clear
  B)
             1 done clear
  C)
             0 done clear
  D)
             a done clear
  View Solution play_arrow

• question_answer166)
  Derivative of \[{{x}^{6}}+{{6}^{x}}\]with respect to x is [Kerala (Engg.) 2002]
  A)
             \[12x\] done clear
  B)
             \[x+4\] done clear
  C)
             \[6{{x}^{5}}+{{6}^{x}}\,\,\log 6\] done clear
  D)
             \[6{{x}^{5}}+x{{6}^{x-1}}\] done clear
  View Solution play_arrow

• question_answer167)
  If \[\sin y+{{e}^{-x\,\cos y}}=e,\]then \[\frac{dy}{dx}\] at \[(1,\pi )\] is [Kerala (Engg.) 2002]
  A)
             \[\sin y\] done clear
  B)
             \[-x\cos y\] done clear
  C)
             \[e\] done clear
  D)
    \[\sin y-x\,\cos y\] done clear
  View Solution play_arrow

• question_answer168)
  The derivative of \[f(x)=3|2+x|\] at the point \[{{x}_{0}}=-3\] is                                                               [Orissa JEE 2002]
  A)
             3 done clear
  B)
             ? 3 done clear
  C)
             0 done clear
  D)
             Does not exist done clear
  View Solution play_arrow

• question_answer169)
  Derivative of the function \[f(x)={{\log }_{5}}({{\log }_{7}}x)\], \[x>7\] is [Orissa JEE 2002]
  A)
             \[\frac{1}{x(\text{In}\,\text{5)(In}\,\text{7)(lo}{{\text{g}}_{\text{7}}}x)}\] done clear
  B)
             \[\frac{1}{x(\text{ln}\,\text{5)(ln}\,\text{7)}}\] done clear
  C)
             \[\frac{1}{x(In\,x)}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer170)
  If \[y={{\cot }^{-1}}({{x}^{2}})\], then \[\frac{dy}{dx}\] is equal to        [Pb. CET 2002]
  A)
             \[\frac{2x}{1+{{x}^{4}}}\] done clear
  B)
             \[\frac{2x}{\sqrt{1+4x}}\] done clear
  C)
             \[\frac{-2x}{1+{{x}^{4}}}\] done clear
  D)
             \[\frac{-2x}{\sqrt{1+{{x}^{2}}}}\] done clear
  View Solution play_arrow

• question_answer171)
  If \[y=\log \tan \sqrt{x}\]then the value of \[\frac{dy}{dx}\] is  [Pb. CET 2000]
  A)
             \[\frac{1}{2\sqrt{x}}\] done clear
  B)
             \[\frac{{{\sec }^{2}}\sqrt{x}}{\sqrt{x}\tan x}\] done clear
  C)
             \[2{{\sec }^{2}}\sqrt{x}\] done clear
  D)
             \[\frac{{{\sec }^{2}}\sqrt{x}}{2\sqrt{x}\tan \sqrt{x}}\] done clear
  View Solution play_arrow

• question_answer172)
  If \[y={{(\cos {{x}^{2}})}^{2}}\]then \[\frac{dy}{dx}\]is equal to            [Pb. CET 2004]
  A)
             \[-4x\sin 2{{x}^{2}}\] done clear
  B)
             \[-x\sin {{x}^{2}}\] done clear
  C)
             \[-2x\sin 2{{x}^{2}}\] done clear
  D)
             \[-x\cos 2{{x}^{2}}\] done clear
  View Solution play_arrow

• question_answer173)
  If \[y={{\tan }^{-1}}(\sec x-\tan x)\]then \[\frac{dy}{dx}=\] [Karnataka CET 2004]
  A)
             2 done clear
  B)
             ?2 done clear
  C)
           ½ done clear
  D)
             ?1/2 done clear
  View Solution play_arrow

• question_answer174)
  If \[y={{\cos }^{-1}}\cos (|x|-f(x)),\] where            \[\]   \[f(x)\left\{ \begin{align}   & =1\,,\,\text{if}\,\,\,x>0 \\  & =-1\,,\,\text{if}\,\,\,x<0 \\  & =0\,,\,\text{if}\,\,\,x=0 \\ \end{align} \right.\],  then \[{{\left. \frac{dy}{dx} \right|}_{x=\frac{5\pi }{4}}}\]is    [J & K 2005]
  A)
             ? 1 done clear
  B)
             1 done clear
  C)
             0 done clear
  D)
             Indeterminate done clear
  View Solution play_arrow

• question_answer175)
  If \[{{x}^{m}}{{y}^{n}}={{(x+y)}^{m+n}}\]then \[{{\left. \frac{dy}{dx} \right|}_{x=1,y=2}}\] is equal to            [J & K 2005]
  A)
             ½ done clear
  B)
             2 done clear
  C)
             2m/n done clear
  D)
             m/ 2n done clear
  View Solution play_arrow

• question_answer176)
  If \[y=\frac{{{e}^{x}}+{{e}^{-x}}}{{{e}^{x}}-{{e}^{-x}}}\] then \[\frac{dy}{dx}\] is equal to    [Karnataka CET 2005]
  A)
             \[\sec {{\text{h}}^{2}}x\] done clear
  B)
             \[\text{cosec}{{\text{h}}^{2}}x\] done clear
  C)
             \[-\sec \,{{\text{h}}^{2}}x\] done clear
  D)
             \[-\text{cosec}{{\text{h}}^{2}}x\] done clear
  View Solution play_arrow

• question_answer177)
  The derivative of function \[f(x)\] is \[{{\tan }^{4}}x\]. If \[f(0)=0\] then \[\underset{x\to 0}{\mathop{\lim }}\,\frac{f(x)}{x}\]is equal to [J & K 2005]
  A)
             1 done clear
  B)
           0 done clear
  C)
             ?1 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer178)
   Let \[f(x)\] be a polynomial function of the second degree. If \[f(1)=f(-1)\] and \[{{a}_{1}},{{a}_{2}},{{a}_{3}}\] are in A.P. then \[{f}'({{a}_{1}})\], \[{f}'({{a}_{2}})\], \[{f}'({{a}_{3}})\] are in [AMU 2005]
  A)
           A.P done clear
  B)
             G.P. done clear
  C)
             H.P. done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer179)
  \[\frac{d}{dx}\,\,\left[ {{\tan }^{-1}}\left( \frac{\sqrt{x}(3-x)}{1-3x} \right) \right]\]=                                                [Kerala (Engg.) 2005]
  A)
             \[\frac{1}{2(1+x)\,\sqrt{x}}\] done clear
  B)
             \[\frac{3}{(1+x)\,\sqrt{x}}\] done clear
  C)
             \[\frac{2}{(1+x)\,\sqrt{x}}\] done clear
  D)
             \[\frac{2\sqrt{2}y-3}{2\sqrt{2}}=\frac{-3\sqrt{2}\times +3}{2\sqrt{2}}\] done clear
  E)
             \[\frac{3}{2(1+x)\sqrt{x}}\] done clear
  View Solution play_arrow

• question_answer180)
   If \[r={{[2\varphi +{{\cos }^{2}}(2\varphi +\pi /4)]}^{1/2}}\] then what is the value of the derivative of \[dr/d\varphi \] at \[\varphi =\pi /4\] [Orissa JEE 2005]
  A)
             \[2\,{{\left( \frac{1}{\pi +1} \right)}^{1/2}}\] done clear
  B)
             \[2\,{{\left( \frac{2}{\pi +1} \right)}^{-1/2}}\] done clear
  C)
             \[2\,{{\left( \frac{1}{\pi +1} \right)}^{-1/2}}\] done clear
  D)
             \[2\,{{\left( \frac{2}{\pi +1} \right)}^{1/2}}\] done clear
  View Solution play_arrow

• question_answer181)
  If \[f(x)=\cos x\cos 2x\cos 4x\cos 8x\cos 16x\], then \[{f}'\left( \frac{\pi }{4} \right)\] is                                                               [AMU 2005]
  A)
             \[\sqrt{2}\] done clear
  B)
             \[\frac{1}{\sqrt{2}}\] done clear
  C)
             1 done clear
  D)
             \[\frac{\sqrt{3}}{2}\] done clear
  View Solution play_arrow

• question_answer182)
  The derivative of \[y=(1-x)\,(2-x)....(n-x)\] at \[x=1\] is equal to                                     [Kerala (Engg.) 2005]
  A)
             0 done clear
  B)
             \[(-1)\,(n-1)\,!\] done clear
  C)
             \[n!\,-\,1\] done clear
  D)
             \[{{(-1)}^{n-1}}(n-1)\,!\] done clear
  E)
             \[{{(-1)}^{n}}\,(n-1)\,!\] done clear
  View Solution play_arrow

• question_answer183)
   If \[y={{\tan }^{-1}}\left( \frac{a\cos x-b\sin x}{b\cos x+a\sin x} \right)\] then \[\frac{dy}{dx}=\] [Kerala (Engg.) 2005]
  A)
             2 done clear
  B)
             ? 1 done clear
  C)
             \[\frac{a}{b}\] done clear
  D)
             0 done clear
  E)
                  \[\frac{b}{a}\] done clear
  View Solution play_arrow