# 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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