JEE Main & Advanced Mathematics Differentiation Some Standard Differentiation

Some Standard Differentiation

Category : JEE Main & Advanced

 

 

 

(1) Differentiation of algebraic functions : \[\frac{d}{dx}{{x}^{n}}=n{{x}^{n-1}}\]

 

 

In particular

 

 

(i)  \[\frac{d}{dx}{{[f(x)]}^{n}}=n\,{{[f(x)]}^{\,n-1}}{f}'(x)\]             

 

 

(ii) \[\frac{d}{dx}(\sqrt{x})=\frac{1}{2\sqrt{x}}\]

 

 

(iii) \[\frac{d}{dx}\left( \frac{1}{{{x}^{n}}} \right)=-\frac{n}{{{x}^{n+1}}}\]

 

 

(2) Differentiation of trigonometric functions :

 

 

(i) \[\frac{d}{dx}\sin x=\cos x\]

 

 

(ii) \[\frac{d}{dx}\cos x=-\sin x\]

 

 

(iii) \[\frac{d}{dx}\tan x={{\sec }^{2}}x\]

 

 

(iv) \[\frac{d}{dx}\sec x=\sec x\tan x\]

 

 

(v) \[\frac{d}{dx}\text{cosec}\,x=-\text{cosec}\,x\,\cot x\]

 

 

(vi) \[\frac{d}{dx}\cot x=-\text{cose}{{\text{c}}^{2}}x\]

 

 

(3) Differentiation of logarithmic and exponential functions : 

 

 

 

(i) \[\frac{d}{dx}\log x=\frac{1}{x}\], for \[x>0\]               

 

 

(ii)  \[\frac{d}{dx}{{e}^{x}}={{e}^{x}}\]  

 

 

(iii) \[\frac{d}{dx}{{a}^{x}}={{a}^{x}}\log a\], for \[a>0\]

 

 

(iv) \[\frac{d}{dx}{{\log }_{a}}x=\frac{1}{x\log a}\], for \[x>0,\,\,a>0,\,\,a\ne 1\]

 

 

(4) Differentiation of inverse trigonometrical functions:

 

 

(i) \[\frac{d}{dx}{{\sin }^{-1}}x=\frac{1}{\sqrt{1-{{x}^{2}}}}\], for \[-1<x<1\]

 

 

(ii)  \[\frac{d}{dx}{{\cos }^{-1}}x=\frac{-1}{\sqrt{1-{{x}^{2}}}}\], for \[-1<x<1\]

 

 

(iii) \[\frac{d}{dx}{{\sec }^{-1}}x=\frac{1}{|x|\sqrt{{{x}^{2}}-1}}\], for \[|x|>1\]

 

 

(iv) \[\frac{d}{dx}\text{cose}{{\text{c}}^{-1}}x=\frac{-1}{|x|\sqrt{{{x}^{2}}-1}}\], for \[|x|>1\]

 

 

(v) \[\frac{d}{dx}{{\tan }^{-1}}x=\frac{1}{1+{{x}^{2}}}\], for \[x\in R\]

 

 

(vi) \[f'(c)\ge 0(f'(c)<0\], for \[x\in R\]

 

 

(5) Differentiation of hyperbolic functions :

 

 

(i) \[\frac{d}{dx}\sinh \,x=\cosh x\]        

 

 

(ii) \[\frac{d}{dx}\cosh \,x=\sinh \,x\]

 

 

(iii) \[\frac{d}{dx}\tanh \,x=\sec {{\text{h}}^{2}}x\]        

 

 

 

(iv) \[\frac{d}{dx}\coth \,x=-\,\text{cosec}{{\text{h}}^{2}}x\]

 

 

(v) \[\frac{d}{dx}\text{sech}\,x=-\text{sech}\,x\tanh x\]             

 

 

(vi) \[\frac{d}{dx}\text{cosech}\,x=-\text{cosech}\,x\,\,\coth x\]

 

 

(vii) \[\frac{d}{dx}{{\sinh }^{-1}}x=1/\sqrt{(1+{{x}^{2}})}\]

 

 

(viii) \[\frac{d}{dx}{{\cosh }^{-1}}x=1/\sqrt{({{x}^{2}}-1)}\]

 

 

(ix) \[\frac{d}{dx}{{\tanh }^{-1}}x=1/({{x}^{2}}-1)\]

 

 

(x) \[\frac{d}{dx}{{\coth }^{-1}}x=1/(1-{{x}^{2}})\]

 

 

(xi) \[\frac{d}{dx}\sec {{\text{h}}^{-1}}x=-1/x\sqrt{(1-{{x}^{2}}})\]

 

 

(xii) \[\frac{d}{dx}\text{cosec}{{\text{h}}^{-1}}x=-1/x\sqrt{(1+{{x}^{2}})}\]

 

 

(6) Suitable substitutions

 

 

Function

Substitution

Function

Substitution

\[\sqrt{{{a}^{2}}-{{x}^{2}}}\]

\[x=a\sin \theta \]

or \[a\cos \theta \]

\[\sqrt{{{x}^{2}}+{{a}^{2}}}\]

\[x=a\tan \theta \]

or \[a\cot \theta \]

\[\sqrt{{{x}^{2}}-{{a}^{2}}}\]

\[x=a\sec \theta \]

or \[a\cos ec\theta \]

\[\sqrt{\frac{a-x}{a+x}}\]

\[x=a\cos 2\theta \]

\[\sqrt{\frac{{{a}^{2}}-{{x}^{2}}}{{{a}^{2}}+{{x}^{2}}}}\]

\[{{x}^{2}}={{a}^{2}}\cos 2\theta \]

\[\sqrt{ax-{{x}^{2}}}\]

\[x=a{{\sin }^{2}}\theta \]

\[\sqrt{\frac{x}{a+x}}\]

\[x=a{{\tan }^{2}}\theta \]

\[\sqrt{\frac{x}{a-x}}\]

\[x=a{{\sin }^{2}}\theta \]

\[\sqrt{(x-a)(x-b)}\]

 

 

\[\begin{align} & x=a{{\sec }^{2}}\theta  \\ & -b{{\tan }^{2}}\theta  \\ \end{align}\]

\[{{f}_{xy}}={{f}_{yx}}\]

\[\begin{align} & x=a{{\cos }^{2}}\theta  \\ & +b{{\sin }^{2}}\theta  \\\end{align}\]

 

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