JEE Main & Advanced Mathematics Differentiation Question Bank

done Differentiation by Substitution Total Questions - 84

• question_answer1)
  If \[y=\sin (2{{\sin }^{-1}}x),\]then \[\frac{dy}{dx}=\]                     [AI CBSE 1983]
  A)
             \[\frac{2-4{{x}^{2}}}{\sqrt{1-{{x}^{2}}}}\] done clear
  B)
             \[\frac{2+4{{x}^{2}}}{\sqrt{1-{{x}^{2}}}}\] done clear
  C)
             \[\frac{2-4{{x}^{2}}}{\sqrt{1+{{x}^{2}}}}\] done clear
  D)
             \[\frac{2+4{{x}^{2}}}{\sqrt{1+{{x}^{2}}}}\] done clear
  View Solution play_arrow

• question_answer2)
  If \[y={{\cos }^{-1}}\left( \frac{3\cos x+4\sin x}{5} \right)\], then \[\frac{dy}{dx}=\]
  A)
             0 done clear
  B)
             1 done clear
  C)
             \[-1\] done clear
  D)
             \[\frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer3)
  \[\frac{d}{dx}{{\cos }^{-1}}\frac{x-{{x}^{-1}}}{x+{{x}^{-1}}}\]=      [DSSE 1985; Rookee 1963]
  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_answer4)
  If \[y={{\sin }^{-1}}\frac{2x}{1+{{x}^{2}}}+{{\sec }^{-1}}\frac{1+{{x}^{2}}}{1-{{x}^{2}}}\], then \[\frac{dy}{dx}\]=            [RPET 1996]
  A)
             \[\frac{4}{1-{{x}^{2}}}\] done clear
  B)
             \[\frac{1}{1+{{x}^{2}}}\] done clear
  C)
             \[\frac{4}{1-{{x}^{2}}}\] done clear
  D)
             \[\frac{-4}{1+{{x}^{2}}}\] done clear
  View Solution play_arrow

• question_answer5)
  If \[y={{\tan }^{-1}}\frac{x}{1+\sqrt{1-{{x}^{2}}}}+\sin \left\{ 2{{\tan }^{-1}}\sqrt{\left( \frac{1-x}{1+x} \right)} \right\}\],then \[\frac{dy}{dx}\]=
  A)
             \[\frac{x}{\sqrt{1-{{x}^{2}}}}\] done clear
  B)
             \[\frac{1-2x}{\sqrt{1-{{x}^{2}}}}\] done clear
  C)
             \[\frac{1-2x}{2\sqrt{1-{{x}^{2}}}}\] done clear
  D)
             \[\frac{1}{1+{{x}^{2}}}\] done clear
  View Solution play_arrow

• question_answer6)
  If \[y={{\sin }^{-1}}\frac{2x}{1+{{x}^{2}}},\]where \[0<x<1\]and \[0<y<\frac{\pi }{2},\] then \[\frac{dy}{dx}=\]  
  A)
             \[\frac{2}{1+{{x}^{2}}}\] done clear
  B)
             \[\frac{2x}{1+{{x}^{2}}}\] done clear
  C)
             \[\frac{-2}{1+{{x}^{2}}}\] done clear
  D)
             \[\frac{-x}{1+{{x}^{2}}}\] done clear
  View Solution play_arrow

• question_answer7)
  \[\frac{d}{dx}{{\tan }^{-1}}\frac{x}{\sqrt{{{a}^{2}}-{{x}^{2}}}}=\]
  A)
             \[\frac{a}{{{a}^{2}}+{{x}^{2}}}\] done clear
  B)
             \[\frac{-a}{{{a}^{2}}+{{x}^{2}}}\] done clear
  C)
             \[\frac{1}{a\sqrt{{{a}^{2}}-{{x}^{2}}}}\] done clear
  D)
             \[\frac{1}{\sqrt{{{a}^{2}}-{{x}^{2}}}}\] done clear
  View Solution play_arrow

• question_answer8)
  \[\frac{d}{dx}{{\cos }^{-1}}\sqrt{\frac{1+{{x}^{2}}}{2}}=\]                                             [AI CBSE 1988]
  A)
             \[\frac{-1}{2\sqrt{1-{{x}^{4}}}}\] done clear
  B)
             \[\frac{1}{2\sqrt{1-{{x}^{4}}}}\] done clear
  C)
             \[\frac{-x}{\sqrt{1-{{x}^{4}}}}\] done clear
  D)
             \[\frac{x}{\sqrt{1-{{x}^{4}}}}\] done clear
  View Solution play_arrow

• question_answer9)
  \[\frac{d}{dx}{{\tan }^{-1}}\left[ \frac{3{{a}^{2}}x-{{x}^{3}}}{a({{a}^{2}}-3{{x}^{2}})} \right]\]at \[x=0\]is
  A)
             \[\frac{1}{a}\] done clear
  B)
             \[\frac{3}{a}\] done clear
  C)
             \[3a\] done clear
  D)
             3 done clear
  View Solution play_arrow

• question_answer10)
  If \[\sqrt{1-{{x}^{2}}}+\sqrt{1-{{y}^{2}}}=a(x-y)\], then \[\frac{dy}{dx}=\] [MNR 1983; ISM Dhanbad 1987; RPET 1991]
  A)
             \[\sqrt{\frac{1-{{x}^{2}}}{1-{{y}^{2}}}}\] done clear
  B)
             \[\sqrt{\frac{1-{{y}^{2}}}{1-{{x}^{2}}}}\] done clear
  C)
             \[\sqrt{\frac{{{x}^{2}}-1}{1-{{y}^{2}}}}\] done clear
  D)
             \[\sqrt{\frac{{{y}^{2}}-1}{1-{{x}^{2}}}}\] done clear
  View Solution play_arrow

• question_answer11)
  \[\frac{d}{dx}{{\sin }^{-1}}(2ax\sqrt{1-{{a}^{2}}{{x}^{2}}})=\]
  A)
             \[\frac{2a}{\sqrt{{{a}^{2}}-{{x}^{2}}}}\] done clear
  B)
             \[\frac{a}{\sqrt{{{a}^{2}}-{{x}^{2}}}}\] done clear
  C)
             \[\frac{2a}{\sqrt{1-{{a}^{2}}{{x}^{2}}}}\] done clear
  D)
             \[\frac{a}{\sqrt{1-{{a}^{2}}{{x}^{2}}}}\] done clear
  View Solution play_arrow

• question_answer12)
  \[\frac{d}{dx}\left\{ {{\cos }^{-1}}\left( \frac{1-{{x}^{2}}}{1+{{x}^{2}}} \right) \right\}=\]                                [AISSE 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_answer13)
  If \[y={{\sin }^{-1}}\sqrt{1-{{x}^{2}}}\], then \[dy/dx=\]                 [AISSE 1987]
  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)
             \[-\frac{1}{\sqrt{{{x}^{2}}-1}}\] done clear
  View Solution play_arrow

• question_answer14)
  The differential coefficient of \[{{\cos }^{-1}}\left\{ \sqrt{\frac{1+x}{2}} \right\}\]with respect to x  is      [MP PET 1993]
  A)
             \[-\frac{1}{2\sqrt{1-{{x}^{2}}}}\] done clear
  B)
             \[\frac{1}{2\sqrt{1-{{x}^{2}}}}\] done clear
  C)
             \[\frac{1}{\sqrt{1-x}}\] done clear
  D)
             \[{{\sin }^{-1}}\left\{ \sqrt{\frac{1+x}{2}} \right\}\] done clear
  View Solution play_arrow

• question_answer15)
  If \[y={{\tan }^{-1}}\sqrt{\frac{a-x}{a+x}}\], then \[\frac{dy}{dx}=\]
  A)
             \[{{\cos }^{-1}}\frac{x}{a}\] done clear
  B)
             \[-{{\cos }^{-1}}\frac{x}{a}\] done clear
  C)
             \[\frac{1}{2}{{\cos }^{-1}}\frac{x}{a}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer16)
  If \[y={{\sin }^{-1}}\frac{\sqrt{(1+x)}+\sqrt{(1-x)}}{2}\], then \[\frac{dy}{dx}=\]
  A)
             \[\frac{1}{\sqrt{(1-{{x}^{2}})}}\] done clear
  B)
             \[-\frac{1}{\sqrt{(1-{{x}^{2}})}}\] done clear
  C)
             \[-\frac{1}{2\sqrt{(1-{{x}^{2}})}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer17)
  If\[f(x)=x+2,\] then \[f'(f(x))\] at x = 4 is                [DCE 2001]
  A)
             8 done clear
  B)
             1 done clear
  C)
             4 done clear
  D)
             5 done clear
  View Solution play_arrow

• question_answer18)
  If \[f(x)={{\cot }^{-1}}\left( \frac{{{x}^{x}}-{{x}^{-x}}}{2} \right)\,,\]then \[f'(1)\] is equal to [RPET 2000]
  A)
             ? 1 done clear
  B)
             1 done clear
  C)
             \[\log \,\,2\] done clear
  D)
             \[-\log \,2\] done clear
  View Solution play_arrow

• question_answer19)
  Let \[3f(x)-2f(1/x)=x,\] then \[f'(2)\]is equal to [MP PET 2000]
  A)
             \[2/7\] done clear
  B)
             \[1/2\] done clear
  C)
             2 done clear
  D)
             \[7/2\] done clear
  View Solution play_arrow

• question_answer20)
  \[\frac{d}{dx}\left[ {{\sin }^{2}}{{\cot }^{-1}}\left\{ \sqrt{\frac{1-x}{1+x}} \right\} \right]\] equals                              [MP PET 2002]
  A)
             \[-1\] done clear
  B)
             \[\frac{1}{2}\] done clear
  C)
             \[-\frac{1}{2}\] done clear
  D)
             1 done clear
  View Solution play_arrow

• question_answer21)
  If \[y={{\tan }^{-1}}\left( \frac{x}{1+\sqrt{1-{{x}^{2}}}} \right)\], then \[\frac{dy}{dx}=\]
  A)
             \[\frac{1}{2\sqrt{1-{{x}^{2}}}}\] done clear
  B)
             \[1-\sqrt{1-{{x}^{2}}}\] done clear
  C)
             \[\frac{1}{2}\] done clear
  D)
             \[\frac{1}{\sqrt{1-{{x}^{2}}}}\] done clear
  View Solution play_arrow

• question_answer22)
  Differential coefficient of \[{{\cos }^{-1}}(\sqrt{x})\]with respect to \[\sqrt{(1-x)}\] is  [MP PET 1997]
  A)
             \[\sqrt{x}\] done clear
  B)
             \[-\sqrt{x}\] done clear
  C)
   \[\frac{1}{\sqrt{x}}\] done clear
  D)
    \[-\frac{1}{\sqrt{x}}\] done clear
  View Solution play_arrow

• question_answer23)
  If \[y={{\tan }^{-1}}\left( \frac{x}{\sqrt{1-{{x}^{2}}}} \right)\], then \[\frac{dy}{dx}=\]      [MP PET 1999]
  A)
    \[-\frac{1}{\sqrt{1-{{x}^{2}}}}\] done clear
  B)
   \[\frac{x}{\sqrt{1-{{x}^{2}}}}\] done clear
  C)
   \[\frac{1}{\sqrt{1-{{x}^{2}}}}\] done clear
  D)
    \[\frac{\sqrt{1-{{x}^{2}}}}{x}\] done clear
  View Solution play_arrow

• question_answer24)
  If \[y={{\sin }^{-1}}\left( \frac{1-{{x}^{2}}}{1+{{x}^{2}}} \right)\], then \[\frac{dy}{dx}\] equals [EAMCET 1991; RPET 1996]
  A)
    \[\frac{2}{1-{{x}^{2}}}\] done clear
  B)
    \[\frac{1}{1+{{x}^{2}}}\] done clear
  C)
             \[\pm \frac{2}{1+{{x}^{2}}}\] done clear
  D)
              \[-\frac{2}{1+{{x}^{2}}}\] done clear
  View Solution play_arrow

• question_answer25)
  The differential coefficient of \[{{\tan }^{-1}}\left( \frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}} \right)\] is [MP PET 2003]
  A)
             \[\sqrt{1-{{x}^{2}}}\] done clear
  B)
             \[\frac{1}{\sqrt{1-{{x}^{2}}}}\] done clear
  C)
             \[\frac{1}{2\sqrt{1-{{x}^{2}}}}\] done clear
  D)
             x done clear
  View Solution play_arrow

• question_answer26)
  \[\frac{d}{dx}\left( {{\tan }^{-1}}\frac{\sqrt{1+{{x}^{2}}}-1}{x} \right)\] is equal to            [MP PET 2004]
  A)
             \[\frac{1}{1+{{x}^{2}}}\] done clear
  B)
             \[\frac{1}{2(1+{{x}^{2}})}\] done clear
  C)
             \[\frac{{{x}^{2}}}{2\sqrt{1+{{x}^{2}}}(\sqrt{1+{{x}^{2}}}-1)}\] done clear
  D)
             \[\frac{2}{1+{{x}^{2}}}\] done clear
  View Solution play_arrow

• question_answer27)
  Differential coefficient of \[{{\sin }^{-1}}\frac{1-x}{1+x}w.r.t\]\[\sqrt{x}\]is          [Roorkee 1984]
  A)
             \[\frac{1}{2\sqrt{x}}\] done clear
  B)
             \[\frac{\sqrt{x}}{\sqrt{1-x}}\] done clear
  C)
             1 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer28)
  The differential coefficient of \[{{\tan }^{-1}}\frac{2x}{1-{{x}^{2}}}\] w.r.t.  \[{{\sin }^{-1}}\frac{2x}{1+{{x}^{2}}}\] is  [Roorkee 1966; BIT Ranchi 1996; Karnataka CET 1994; MP PET 1999; UPSEAT 1999, 2001]
  A)
             1 done clear
  B)
             ? 1 done clear
  C)
             0 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer29)
  Differential coefficient of\[{{\sin }^{-1}}x\] w.r.t \[{{\cos }^{-1}}\sqrt{1-{{x}^{2}}}\] is  [MNR 1983; AMU 2002]
  A)
             1 done clear
  B)
             \[\frac{1}{1+{{x}^{2}}}\] done clear
  C)
             2 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer30)
  Differential coefficient of \[\frac{{{\tan }^{-1}}x}{1+{{\tan }^{-1}}x}\] w.r.t. \[{{\tan }^{-1}}x\] is
  A)
             \[\frac{1}{1+{{\tan }^{-1}}x}\] done clear
  B)
             \[\frac{-1}{1+{{\tan }^{-1}}x}\] done clear
  C)
             \[\frac{1}{{{(1+{{\tan }^{-1}}x)}^{^{2}}}}\] done clear
  D)
             \[\frac{-1}{2\,{{(1+{{\tan }^{-1}}x)}^{2}}}\] done clear
  View Solution play_arrow

• question_answer31)
  The differential coefficient of \[{{\tan }^{-1}}\left( \frac{\sqrt{1+{{x}^{2}}}-1}{x} \right)\] with respect to \[{{\tan }^{-1}}\]x  is   [Kurukshetra CEE 1998; RPET 1999]
  A)
             \[\frac{1}{2}\] done clear
  B)
             \[-\frac{1}{2}\] done clear
  C)
             1 done clear
  D)
   None of these done clear
  View Solution play_arrow

• question_answer32)
  The differential coefficient of \[{{\tan }^{-1}}\sqrt{x}\] with respect to \[\sqrt{x}\] is       [MP PET 1987]
  A)
             \[\frac{1}{\sqrt{1+x}}\] done clear
  B)
             \[\frac{1}{2x\sqrt{1+x}}\] done clear
  C)
             \[\frac{1}{2\sqrt{x(1+x)}}\] done clear
  D)
             \[\frac{1}{1+x}\] done clear
  View Solution play_arrow

• question_answer33)
  Differential coefficient of \[{{x}^{3}}\] with respect to \[{{x}^{2}}\] is [RPET 1995]
  A)
   \[\]\[\frac{3{{x}^{2}}}{2}\] done clear
  B)
    \[\frac{3x}{2}\] done clear
  C)
    \[\frac{3{{x}^{3}}}{2}\] done clear
  D)
   \[\frac{3}{2x}\] done clear
  View Solution play_arrow

• question_answer34)
  The differential coefficient of  \[{{x}^{6}}\] with respect to \[{{x}^{3}}\] is                                  [EAMCET 1988; UPSEAT 2000]
  A)
             \[5{{x}^{2}}\] done clear
  B)
             \[3{{x}^{3}}\] done clear
  C)
             \[5{{x}^{5}}\] done clear
  D)
             \[2{{x}^{3}}\] done clear
  View Solution play_arrow

• question_answer35)
  Derivative of \[{{\sec }^{-1}}\left\{ \frac{1}{2{{x}^{2}}-1} \right\}\]w.r.t \[\sqrt{1+3x}\]at \[x=-\frac{1}{3}\] is                          [EAMCET 1991]
  A)
             0 done clear
  B)
             1/2 done clear
  C)
             1/3 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer36)
  Differential coefficient of \[{{\tan }^{-1}}\sqrt{\frac{1-{{x}^{2}}}{1+{{x}^{2}}}}\] w.r.t. \[{{\cos }^{-1}}({{x}^{2}})\] is                     [RPET 1996]
  A)
   \[\frac{1}{2}\] done clear
  B)
             \[-\frac{1}{2}\] done clear
  C)
             1 done clear
  D)
             0 done clear
  View Solution play_arrow

• question_answer37)
  If  \[u={{\tan }^{-1}}\left\{ \frac{\sqrt{1+{{x}^{2}}}-1}{x} \right\}\] and \[v=2{{\tan }^{-1}}x\], then \[\frac{du}{dv}\]  is equal to                                                      [RPET 1997]
  A)
             4 done clear
  B)
             1 done clear
  C)
             ¼ done clear
  D)
             ?1/4 done clear
  View Solution play_arrow

• question_answer38)
  The derivative of \[{{\sin }^{-1}}\left( \frac{2x}{1+{{x}^{2}}} \right)\,\]w.r.t. \[{{\cos }^{-1}}\left( \frac{1-{{x}^{2}}}{1+{{x}^{2}}} \right)\] is    [Karnataka CET 2000; Pb. CET 2004]
  A)
             ?1 done clear
  B)
             1 done clear
  C)
             2 done clear
  D)
             4 done clear
  View Solution play_arrow

• question_answer39)
  The derivative of \[{{\sin }^{2}}x\]with respect to \[{{\cos }^{2}}x\] is  [DCE 2002]
  A)
             \[{{\tan }^{2}}x\] done clear
  B)
             \[\tan x\] done clear
  C)
             \[-\tan x\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer40)
  Differential coefficient of \[{{\tan }^{-1}}\left( \frac{x}{1+\sqrt{1-{{x}^{2}}}} \right)\] w.r.t \[{{\sin }^{-1}}x,\] is [Kurukshetra CEE 2002]
  A)
             \[\frac{1}{2}\] done clear
  B)
             1 done clear
  C)
             2 done clear
  D)
             \[\frac{3}{2}\] done clear
  View Solution play_arrow

• question_answer41)
  The derivative of \[{{\cos }^{-1}}\left( \frac{1-{{x}^{2}}}{1+{{x}^{2}}} \right)\] w.r.t. \[{{\cot }^{-1}}\left( \frac{1-3{{x}^{2}}}{3x-{{x}^{2}}} \right)\] is                                                               [Karnataka CET 2003]
  A)
             1 done clear
  B)
             \[\frac{3}{2}\] done clear
  C)
             \[\frac{2}{3}\] done clear
  D)
             \[\frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer42)
  The differential of \[{{e}^{{{x}^{3}}}}\]with respect to \[\log x\] is [Karnataka CET 2002]
  A)
             \[{{e}^{{{x}^{3}}}}\] done clear
  B)
             \[3{{x}^{2}}{{e}^{{{x}^{3}}}}\] done clear
  C)
             \[3{{x}^{3}}{{e}^{{{x}^{3}}}}\] done clear
  D)
             \[3{{x}^{2}}{{e}^{{{x}^{3}}}}+3{{x}^{2}}\] done clear
  View Solution play_arrow

• question_answer43)
  The 2nd derivative of \[a{{\sin }^{3}}t\] with respect to \[a{{\cos }^{3}}t\,\,\text{at}\,\,t=\frac{\pi }{4}\]  is                                [Kerala (Engg.) 2002]
  A)
             \[\frac{4\sqrt{2}}{3a}\] done clear
  B)
             2 done clear
  C)
             \[\frac{1}{12a}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer44)
  The differential coefficient of \[f(\sin x)\] with respect to x, where \[f(x)=\log x\], is                                       [Karnataka CET 2004]
  A)
             \[\tan x\] done clear
  B)
             \[\cot x\] done clear
  C)
             \[f(\cos x)\] done clear
  D)
             \[1/x\] done clear
  View Solution play_arrow

• question_answer45)
  If \[y=\sin x\sin 3x,\]then \[{{y}_{n}}=\]
  A)
             \[\frac{1}{2}\left[ \cos \left( 2x+n\frac{\pi }{2} \right)-\cos \left( 4x+n\frac{\pi }{2} \right) \right]\] done clear
  B)
             \[\frac{1}{2}\left[ {{2}^{n\,\,}}\cos \left( 2x+n\frac{\pi }{2} \right)-{{4}^{n}}\cos \left( 4x+n\frac{\pi }{2} \right) \right]\] done clear
  C)
             \[\frac{1}{2}\left[ {{4}^{n}}\cos \left( 4x+n\frac{\pi }{2} \right)-{{2}^{n}}\cos \left( 2x+n\frac{\pi }{2} \right) \right]\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer46)
  \[{{n}^{th}}\]derivative of \[{{x}^{n+1}}\]is                          
  A)
             \[(n+1)!x\] done clear
  B)
             \[(n+1)!\] done clear
  C)
             \[n!x\] done clear
  D)
             \[n!\] done clear
  View Solution play_arrow

• question_answer47)
  If \[y={{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+.....+{{a}_{n}}{{x}^{n}},\]then \[{{y}_{n}}=\]
  A)
             \[n!\] done clear
  B)
             \[n!{{a}_{n}}x\] done clear
  C)
             \[n!{{a}_{n}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer48)
  If \[y=A\cos nx+B\sin nx,\] then \[\frac{{{d}^{2}}y}{d{{x}^{2}}}=\] [Karnataka CET 1996]
  A)
             \[{{n}^{2}}y\]                        done clear
  B)
             \[-y\] done clear
  C)
             \[-{{n}^{2}}y\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer49)
  \[\frac{{{d}^{n}}}{d{{x}^{n}}}({{e}^{2x}}+{{e}^{-2x}})=\]               
  A)
             \[{{e}^{2x}}+{{(-1)}^{n}}{{e}^{-2x}}\] done clear
  B)
             \[{{2}^{n}}({{e}^{2x}}-{{e}^{-2x}})\] done clear
  C)
             \[{{2}^{n}}[{{e}^{2x}}+{{(-1)}^{n}}{{e}^{-2x}}]\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer50)
  If \[x=\log p\]and \[y=\frac{1}{p}\], then
  A)
             \[\frac{{{d}^{2}}y}{d{{x}^{2}}}-2p=0\] done clear
  B)
             \[\frac{{{d}^{2}}y}{d{{x}^{2}}}+y=0\] done clear
  C)
             \[\frac{{{d}^{2}}y}{d{{x}^{2}}}+\frac{dy}{dx}=0\] done clear
  D)
             \[\frac{{{d}^{2}}y}{d{{x}^{2}}}-\frac{dy}{dx}=0\] done clear
  View Solution play_arrow

• question_answer51)
  If \[f(x)=a\sin (\log x)\], then \[{{x}^{2}}f''(x)+xf'(x)=\]
  A)
             \[f(x)\] done clear
  B)
             \[-f(x)\] done clear
  C)
             0 done clear
  D)
             1 done clear
  View Solution play_arrow

• question_answer52)
  If \[y={{e}^{{{\tan }^{-1}}x}}\], then \[(1+{{x}^{2}})\frac{{{d}^{2}}y}{d{{x}^{2}}}=\]
  A)
             \[(1-2x)\frac{dy}{dx}\] done clear
  B)
             \[-2x\frac{dy}{dx}\] done clear
  C)
             \[-x\frac{dy}{dx}\] done clear
  D)
             0 done clear
  View Solution play_arrow

• question_answer53)
  If \[y={{x}^{2}}{{e}^{mx}}\], where m is a constant, then \[\frac{{{d}^{3}}y}{d{{x}^{3}}}=\] [MP PET 1987]
  A)
             \[m{{e}^{mx}}({{m}^{2}}{{x}^{2}}+6mx+6)\] done clear
  B)
             \[2{{m}^{3}}x{{e}^{mx}}\] done clear
  C)
             \[m{{e}^{mx}}({{m}^{2}}{{x}^{2}}+2mx+2)\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer54)
  If \[y=a{{e}^{mx}}+b{{e}^{-mx}}\], then \[\frac{{{d}^{2}}y}{d{{x}^{2}}}-{{m}^{2}}y=\]     [MP PET 1987]
  A)
             \[{{m}^{2}}(a{{e}^{mx}}-b{{e}^{-mx}})\] done clear
  B)
             1 done clear
  C)
             0 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer55)
  If \[y={{({{x}^{2}}-1)}^{m}}\], then the \[{{(2m)}^{th}}\]differential coefficient of y is                                                     [MP PET 1987]
  A)
             m done clear
  B)
             \[(2m)!\] done clear
  C)
             2m done clear
  D)
             m! done clear
  View Solution play_arrow

• question_answer56)
  If \[y=a{{x}^{n+1}}+b{{x}^{-n}}\], then \[{{x}^{2}}\frac{{{d}^{2}}y}{d{{x}^{2}}}=\]             [Karnataka CET 1993]
  A)
             \[n\,(n-1)y\] done clear
  B)
             \[n\,(n+1)y\] done clear
  C)
             ny done clear
  D)
             \[{{n}^{2}}y\] done clear
  View Solution play_arrow

• question_answer57)
  If \[y=a+b{{x}^{2}};a,b\] arbitrary constants, then [EAMCET 1994]
  A)
             \[\frac{{{d}^{2}}y}{d{{x}^{2}}}=2xy\] done clear
  B)
             \[x\frac{{{d}^{2}}y}{d{{x}^{2}}}=\frac{dy}{dx}\] done clear
  C)
             \[x\frac{{{d}^{2}}y}{d{{x}^{2}}}-\frac{dy}{dx}+y=0\] done clear
  D)
             \[x\frac{{{d}^{2}}y}{d{{x}^{2}}}=2xy\] done clear
  View Solution play_arrow

• question_answer58)
   \[\frac{{{d}^{20}}y}{d{{x}^{20}}}(2\cos x\cos 3x)\]=                                        [EAMCET 1994]
  A)
             \[{{2}^{20}}(\cos 2x-{{2}^{20}}\cos 4x)\] done clear
  B)
             \[{{2}^{20}}(\cos 2x+{{2}^{20}}\cos 4x)\] done clear
  C)
             \[{{2}^{20}}(\sin 2x+{{2}^{20}}\sin 4x)\] done clear
  D)
             \[{{2}^{20}}(\sin 2x-{{2}^{20}}\sin 4x)\] done clear
  View Solution play_arrow

• question_answer59)
  If \[y={{\sin }^{2}}\alpha +{{\cos }^{2}}(\alpha +\beta )+2\sin \alpha \sin \beta \cos (\alpha +\beta )\], then \[\frac{{{d}^{3}}y}{d{{\alpha }^{3}}}\] is, (keeping \[\beta \]as constant)
  A)
             \[\frac{{{\sin }^{3}}(\alpha +\beta )}{\cos \alpha }\] done clear
  B)
             \[\cos (\alpha +3\beta )\] done clear
  C)
             0 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer60)
  If \[y=x\log \left( \frac{x}{a+bx} \right)\], then \[{{x}^{3}}\frac{{{d}^{2}}y}{d{{x}^{2}}}=\] [WB JEE 1991; Roorkee 1976]
  A)
             \[x\frac{dy}{dx}-y\] done clear
  B)
             \[{{\left( x\frac{dy}{dx}-y \right)}^{2}}\] done clear
  C)
             \[y\frac{dy}{dx}-x\] done clear
  D)
             \[{{\left( y\frac{dy}{dx}-x \right)}^{2}}\] done clear
  View Solution play_arrow

• question_answer61)
  If \[{{e}^{y}}+xy=e\], then the value of \[\frac{{{d}^{2}}y}{d{{x}^{2}}}\]for \[x=0\], is  [Kurukshetra CEE 2002]
  A)
             \[\frac{1}{e}\] done clear
  B)
             \[\frac{1}{{{e}^{2}}}\] done clear
  C)
             \[\frac{1}{{{e}^{3}}}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer62)
  If f be a polynomial, then the second derivative of \[f({{e}^{x}})\] is                   [Karnataka CET 1999]
  A)
             \[{f}'({{e}^{x}})\] done clear
  B)
             \[{f}''\,({{e}^{x}})\,{{e}^{x}}+{f}'({{e}^{x}})\] done clear
  C)
             \[{f}''\,({{e}^{x}}){{e}^{2x}}+{f}''({{e}^{x}})\] done clear
  D)
             \[{f}''\,({{e}^{x}}){{e}^{2x}}+{f}'\,({{e}^{x}})\,{{e}^{x}}\] done clear
  View Solution play_arrow

• question_answer63)
  If \[y=\sin x+{{e}^{x}},\]then \[\frac{{{d}^{2}}x}{d{{y}^{2}}}=\]                                  [Karnataka CET 1999; UPSEAT 2001; Kurukshetra CEE 2002]
  A)
             \[{{(-\sin x+{{e}^{x}})}^{-1}}\] done clear
  B)
             \[\frac{\sin x-{{e}^{x}}}{{{(\cos x+{{e}^{x}})}^{2}}}\] done clear
  C)
             \[\frac{\sin x-{{e}^{x}}}{{{(\cos x+{{e}^{x}})}^{3}}}\] done clear
  D)
             \[\frac{\sin x+{{e}^{x}}}{{{(\cos x+{{e}^{x}})}^{3}}}\] done clear
  View Solution play_arrow

• question_answer64)
  If \[y={{x}^{3}}\log {{\log }_{e}}(1+x)\], then \[{y}''\,(0)\] equals [AMU 1999]
  A)
             0 done clear
  B)
             ? 1 done clear
  C)
             \[6\,\,\log {{}_{e}}\,2\] done clear
  D)
             6 done clear
  View Solution play_arrow

• question_answer65)
  \[\frac{{{d}^{2}}x}{d{{y}^{2}}}\] is equal to [AMU 2001]
  A)
             \[\frac{1}{{{(dy/dx)}^{2}}}\] done clear
  B)
             \[\frac{\left( {{d}^{2}}y/d{{x}^{2}} \right)}{{{\left( dy/dx \right)}^{2}}}\] done clear
  C)
             \[\frac{{{d}^{2}}y}{d{{x}^{2}}}\] done clear
  D)
             \[\frac{\left( -{{d}^{2}}y/d{{x}^{2}} \right)}{{{\left( dy/dx \right)}^{2}}}\] done clear
  View Solution play_arrow

• question_answer66)
  A curve is given by the equations  \[x=a\cos \theta +\frac{1}{2}b\cos 2\theta ,\] \[y=a\sin \theta +\frac{1}{2}b\,\sin \,2\theta \], then the points for which \[\frac{{{d}^{2}}y}{d{{x}^{2}}}=0,\] is given by [Kurukshetra CEE 2002]
  A)
             \[\sin \theta =\frac{2{{a}^{2}}+{{b}^{2}}}{5ab}\] done clear
  B)
             \[\tan \theta =\frac{3{{a}^{2}}+2{{b}^{2}}}{4ab}\] done clear
  C)
             \[\cos \theta =\frac{-\left( {{a}^{2}}+2{{b}^{2}} \right)}{3ab}\] done clear
  D)
             \[\cos \theta =\frac{\left( {{a}^{2}}-2{{b}^{2}} \right)}{3ab}\] done clear
  View Solution play_arrow

• question_answer67)
  If \[y={{\left( x+\sqrt{1+{{x}^{2}}} \right)}^{n}},\] then \[(1+{{x}^{2}})\frac{{{d}^{2}}y}{d{{x}^{2}}}+x\frac{dy}{dx}\] is [AIEEE 2002]
  A)
             \[{{n}^{2}}y\] done clear
  B)
             \[-{{n}^{2}}y\] done clear
  C)
             \[-y\] done clear
  D)
             \[2{{x}^{2}}y\] done clear
  View Solution play_arrow

• question_answer68)
  \[f(x)\] and \[g(x)\] are two differentiable function on \[[0,\,2]\] such that \[f''(x)-g''(x)=0,f'(1)=2,g'(1)=4\], \[f(2)=3\], \[g(2)=9,\] then \[f(x)-g(x)\] at \[x=3/2\] is  [AIEEE 2002]
  A)
             0 done clear
  B)
             2 done clear
  C)
             10 done clear
  D)
             ? 5 done clear
  View Solution play_arrow

• question_answer69)
  If \[y=a{{e}^{x}}+b{{e}^{-x}}+c\] where \[a,b,c\] are parameters then \[{{y}''}'=\]                                            [EAMCET 2002]
  A)
             \[y\] done clear
  B)
             \[y'\] done clear
  C)
             0 done clear
  D)
             \[y''\] done clear
  View Solution play_arrow

• question_answer70)
  If \[y=a\cos \,(\log x)+b\sin \,(\log x)\] where \[a,\,b\] are parameters then \[{{x}^{2}}{y}''\,+\,x{y}'\,=\] [EAMCET 2002]
  A)
             \[y\] done clear
  B)
             \[-y\] done clear
  C)
             \[2y\] done clear
  D)
             \[-2y\] done clear
  View Solution play_arrow

• question_answer71)
  If \[u={{x}^{2}}+{{y}^{2}}\] and \[x=s+3t,\]\[y=2s-t,\] then \[\frac{{{d}^{2}}u}{d{{s}^{2}}}=\]                    [Orissa JEE 2002]
  A)
             12 done clear
  B)
             32 done clear
  C)
             36 done clear
  D)
             10 done clear
  View Solution play_arrow

• question_answer72)
  \[\frac{{{d}^{n}}}{d{{x}^{n}}}(\log x)\]=                                                                 [RPET 2002]
  A)
             \[\frac{(n-1)!}{{{x}^{n}}}\] done clear
  B)
             \[\frac{n\,!}{{{x}^{n}}}\] done clear
  C)
             \[\frac{(n-2)!}{{{x}^{n}}}\] done clear
  D)
             \[{{(-1)}^{n-1}}\frac{(n-1)!}{{{x}^{n}}}\] done clear
  View Solution play_arrow

• question_answer73)
  The nth derivative of \[x{{e}^{x}}\] vanishes when [AMU 1999]
  A)
             \[x=0\] done clear
  B)
             \[x=-1\] done clear
  C)
             \[x=-n\] done clear
  D)
             \[x=n\] done clear
  View Solution play_arrow

• question_answer74)
  \[\frac{{{d}^{2}}}{d{{x}^{2}}}(2\cos x\,\cos 3x)=\]                                                                  [RPET 2003]
  A)
             \[{{2}^{2}}(\cos 2x+{{2}^{2}}\cos 4x)\] done clear
  B)
             \[{{2}^{2}}(\cos 2x-{{2}^{2}}\cos 4x)\] done clear
  C)
             \[{{2}^{2}}(-\cos 2x+{{2}^{2}}\cos 4x)\] done clear
  D)
             \[-{{2}^{2}}(\cos 2x+{{2}^{2}}\cos 4x)\] done clear
  View Solution play_arrow

• question_answer75)
  If \[y=1-x+\frac{{{x}^{2}}}{2!}-\frac{{{x}^{3}}}{3!}+\frac{{{x}^{4}}}{4!}-\]....., then \[\frac{{{d}^{2}}y}{d{{x}^{2}}}=\] [Karnataka CET 2003]
  A)
             \[x\] done clear
  B)
             \[-x\] done clear
  C)
             \[-y\] done clear
  D)
             \[y\] done clear
  View Solution play_arrow

• question_answer76)
  If \[f(x)\] is a differentiable function and \[{f}''(0)=a\] then \[\underset{x\to 0}{\mathop{\lim }}\,\frac{2f(x)-3f(2x)+f(4x)}{{{x}^{2}}}\] is                                                            [Orissa JEE 2004]
  A)
             3a done clear
  B)
             2a done clear
  C)
             5a done clear
  D)
             4a done clear
  View Solution play_arrow

• question_answer77)
  If \[x=A\cos 4t+B\sin 4t\],then \[\frac{{{d}^{2}}x}{d{{t}^{2}}}=\] [Karnataka CET 2004]
  A)
             ? 16x done clear
  B)
             16 x done clear
  C)
             x done clear
  D)
             ? x done clear
  View Solution play_arrow

• question_answer78)
  If \[f(x)={{\tan }^{-1}}\left\{ \frac{\log \left( \frac{e}{{{x}^{2}}} \right)}{\log (e{{x}^{2}})} \right\}+{{\tan }^{-1}}\left( \frac{3+2\log x}{1-6\log x} \right)\], then \[\frac{{{d}^{n}}y}{d{{x}^{n}}}\] is \[(n\ge 1)\]
  A)
             \[{{\tan }^{-1}}\{{{(\log x)}^{n}}\}\] done clear
  B)
             0 done clear
  C)
             \[\frac{1}{2}\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer79)
  If \[{{f}_{n}}(x)\], \[{{g}_{n}}(x)\], \[{{h}_{n}}(x),n=1,\,2,\,3\]are polynomials in x such that \[{{f}_{n}}(a)={{g}_{n}}(a)={{h}_{n}}(a),n=1,2,3\] and \[F(x)=\left| \begin{matrix}    {{f}_{1}}(x) & {{f}_{2}}(x) & {{f}_{3}}(x)  \\    {{g}_{1}}(x) & {{g}_{2}}(x) & {{g}_{3}}(x)  \\    {{h}_{1}}(x) & {{h}_{2}}(x) & {{h}_{3}}(x)  \\ \end{matrix} \right|\].                Then \[{F}'(a)\]is equal to
  A)
             0 done clear
  B)
             \[{{f}_{1}}(a){{g}_{2}}(a){{h}_{3}}(a)\] done clear
  C)
             1 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer80)
  Let \[f(x)=\left| \begin{matrix}    {{x}^{3}} & \sin x & \cos x  \\    6 & -1 & 0  \\    p & {{p}^{2}} & {{p}^{3}}  \\ \end{matrix} \right|\], where p is a constant. Then \[\frac{{{d}^{3}}}{d{{x}^{3}}}\left\{ f(x) \right\}\]at \[x=0\]is           [IIT 1997 Cancelled]
  A)
             p done clear
  B)
             \[p+{{p}^{2}}\] done clear
  C)
             \[p+{{p}^{3}}\] done clear
  D)
             Independent of p done clear
  View Solution play_arrow

• question_answer81)
  \[f(x)=\left| \begin{matrix}    {{x}^{3}} & {{x}^{2}} & 3{{x}^{2}}  \\    1 & -6 & 4  \\    p & {{p}^{2}} & {{p}^{3}}  \\ \end{matrix} \right|\] , here p is a constant, then \[\frac{{{d}^{3}}f(x)}{d{{x}^{3}}}\] is            [DCE 2000]
  A)
             Proportional to \[{{x}^{2}}\] done clear
  B)
             Proportional to x done clear
  C)
             Proportional to \[{{x}^{3}}\] done clear
  D)
             A constant done clear
  View Solution play_arrow

• question_answer82)
  If \[y=\sin px\] and \[{{y}_{n}}\] is the nth derivative of y, then \[\left| \,\begin{matrix}    y & {{y}_{1}} & {{y}_{2}}  \\    {{y}_{3}} & {{y}_{4}} & {{y}_{5}}  \\    {{y}_{6}} & {{y}_{7}} & {{y}_{8}}  \\ \end{matrix}\, \right|\] is equal to [AMU 2002]
  A)
              1 done clear
  B)
              0 done clear
  C)
             ? 1 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer83)
   If \[{{y}^{2}}=a{{x}^{2}}+bx+c\], then \[{{y}^{3}}\frac{{{d}^{2}}y}{d{{x}^{2}}}\]is               [DCE 1999]
  A)
             A constant done clear
  B)
             A function of x only done clear
  C)
             A function of y only done clear
  D)
             A function of x and y done clear
  View Solution play_arrow

• question_answer84)
   If \[y={{a}^{x}}.{{b}^{2x-1}}\], then \[\frac{{{d}^{2}}y}{d{{x}^{2}}}\] is                                  [Kerala (Engg.) 2005]
  A)
             \[{{y}^{2}}.\log a{{b}^{2}}\] done clear
  B)
             \[y.\log a{{b}^{2}}\] done clear
  C)
             \[{{y}^{2}}\] done clear
  D)
             \[y.{{(\log {{a}^{2}}b)}^{2}}\] done clear
  E)
                  \[y.{{(\log a{{b}^{2}})}^{2}}\] done clear
  View Solution play_arrow