\displaystyle 1.\ \text{Let }f(x)\text{ be a differentiable or derivable function on }[a,b].\text{ Then,}

\displaystyle \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}\ \text{or,}\ \lim_{h\to 0}\frac{f(x-h)-f(x)}{-h}\qquad (i)

\displaystyle \text{is called the derivative or differentiation of }f(x)\text{ with respect to }x\text{ and is denoted by}

\displaystyle f'(x)\ \text{or,}\ \frac{d}{dx}(f(x))\ \text{or,}\ Df(x),\ \text{where }D=\frac{d}{dx}

 

\displaystyle 2.\ \text{If }y=f(x),\text{ then }\left(\frac{dy}{dx}\right)_{P}\text{ gives the slope of the tangent to the curve }y=f(x)\text{ at point }P.

\displaystyle 3.\ \text{Following are derivatives of some standard functions:}
\displaystyle (i)\ \frac{d}{dx}(x^{n})=nx^{n-1}

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

\displaystyle (iii)\ \frac{d}{dx}(a^{x})=a^{x}\log_{e}a

\displaystyle (iv)\ \frac{d}{dx}(\log x)=\frac{1}{x}

\displaystyle (v)\ \frac{d}{dx}(\log_{a}x)=\frac{1}{x\log_{e}a}

\displaystyle (vi)\ \frac{d}{dx}(\sin x)=\cos x

\displaystyle (vii)\ \frac{d}{dx}(\cos x)=-\sin x

\displaystyle (viii)\ \frac{d}{dx}(\tan x)=\sec^{2}x

\displaystyle (ix)\ \frac{d}{dx}(\cot x)=-\mathrm{cosec}^{2}x

\displaystyle (x)\ \frac{d}{dx}(\sec x)=\sec x\tan x

\displaystyle (xi)\ \frac{d}{dx}(\mathrm{cosec}\ x)=-\mathrm{cosec}\ x\cot x

\displaystyle (xii)\ \frac{d}{dx}(\sin^{-1}x)=\frac{1}{\sqrt{1-x^{2}}},\ -1<x<1

\displaystyle (xiii)\ \frac{d}{dx}(\cos^{-1}x)=-\frac{1}{\sqrt{1-x^{2}}},\ -1<x<1

\displaystyle (xiv)\ \frac{d}{dx}(\tan^{-1}x)=\frac{1}{1+x^{2}},\ -\infty<x<\infty

\displaystyle (xv)\ \frac{d}{dx}(\cot^{-1}x)=-\frac{1}{1+x^{2}},\ -\infty<x<\infty

\displaystyle (xvi)\ \frac{d}{dx}(\sec^{-1}x)=\frac{1}{|x|\sqrt{x^{2}-1}},\ |x|>1

\displaystyle (xvii)\ \frac{d}{dx}(\mathrm{cosec}^{-1}x)=-\frac{1}{|x|\sqrt{x^{2}-1}},\ |x|>1

\displaystyle (xviii)\ \frac{d}{dx}\left\{\sin^{-1}\left(\frac{2x}{1+x^{2}}\right)\right\}=\begin{cases} \frac{2}{1+x^{2}},\ x>1\\ \frac{2}{1+x^{2}},\ -1<x<1\\ -\frac{2}{1+x^{2}},\ x<-1 \end{cases}

\displaystyle (xix)\ \frac{d}{dx}\left\{\cos^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)\right\}=\begin{cases} \frac{2}{1+x^{2}},\ x>0\\ -\frac{2}{1+x^{2}},\ x<0 \end{cases}

\displaystyle (xx)\ \frac{d}{dx}\left\{\tan^{-1}\left(\frac{2x}{1-x^{2}}\right)\right\}=\begin{cases} \frac{2}{1+x^{2}},\ x<-1\ \text{or}\ x>1\\ -\frac{2}{1+x^{2}},\ -1<x<1 \end{cases}

\displaystyle (xxi)\ \frac{d}{dx}\left\{\sin^{-1}(3x-4x^{3})\right\}=\begin{cases} -\frac{3}{\sqrt{1-x^{2}}},\ \frac{1}{2}<x<1\ \text{or}\ -1<x<-\frac{1}{2}\\ \frac{3}{\sqrt{1-x^{2}}},\ -\frac{1}{2}<x<\frac{1}{2} \end{cases}

\displaystyle (xxii)\ \frac{d}{dx}\left\{\cos^{-1}(4x^{3}-3x)\right\}=\begin{cases} -\frac{3}{\sqrt{1-x^{2}}},\ \frac{1}{2}<x<1\\ \frac{3}{\sqrt{1-x^{2}}},\ -\frac{1}{2}<x<\frac{1}{2}\ \text{or}\ -1<x<-\frac{1}{2} \end{cases}

\displaystyle (xxiii)\ \frac{d}{dx}\left\{\tan^{-1}\left(\frac{3x-x^{3}}{1-3x^{2}}\right)\right\}=\begin{cases} \frac{3}{1+x^{2}},\ x<-\frac{1}{\sqrt{3}}\ \text{or}\ x>\frac{1}{\sqrt{3}}\\ -\frac{3}{1+x^{2}},\ -\frac{1}{\sqrt{3}}<x<\frac{1}{\sqrt{3}} \end{cases}

\displaystyle (xxiv)\ \frac{d}{dx}\{\sin(\sin^{-1}x)\}=1,\ -1<x<1

\displaystyle (xxv)\ \frac{d}{dx}\{\cos(\cos^{-1}x)\}=1,\ -1<x<1

\displaystyle (xxvi)\ \frac{d}{dx}\{\tan(\tan^{-1}x)\}=1\ \text{for all }x\in R

\displaystyle (xxvii)\ \frac{d}{dx}\{\mathrm{cosec}(\mathrm{cosec}^{-1}x)\}=1\ \text{for all }x\in R-(-1,1)

\displaystyle (xxviii)\ \frac{d}{dx}\{\sec(\sec^{-1}x)\}=1\ \text{for all }x\in R-(-1,1)

\displaystyle (xxix)\ \frac{d}{dx}\{\cot(\cot^{-1}x)\}=1\ \text{for all }x\in R

\displaystyle (xxx)\ \frac{d}{dx}\{\sin^{-1}(\sin x)\}=\begin{cases} -1,\ -\frac{3\pi}{2}<x<-\frac{\pi}{2}\\ 1,\ -\frac{\pi}{2}<x<\frac{\pi}{2}\\ -1,\ \frac{\pi}{2}<x<\frac{3\pi}{2}\\ 1,\ \frac{3\pi}{2}<x<\frac{5\pi}{2} \end{cases}\ \text{and so on}

\displaystyle (xxxi)\ \frac{d}{dx}\{\cos^{-1}(\cos x)\}=\begin{cases} 1,\ 0<x<\pi\\ -1,\ \pi<x<2\pi \end{cases}\ \text{and so on}

\displaystyle (xxxii)\ \frac{d}{dx}\{\tan^{-1}(\tan x)\}=\{1,\ n\pi-\frac{\pi}{2}<x<n\pi+\frac{\pi}{2},\ n\in Z\}

\displaystyle (xxxiii)\ \frac{d}{dx}\{\mathrm{cosec}^{-1}(\mathrm{cosec}\ x)\}=\begin{cases} 1,\ -\frac{\pi}{2}<x<0\ \text{or}\ 0<x<\frac{\pi}{2}\\ -1,\ \frac{\pi}{2}<x<\pi\ \text{or}\ \pi<x<\frac{3\pi}{2} \end{cases}\ \text{and so on}

\displaystyle (xxxiv)\ \frac{d}{dx}\{\sec^{-1}(\sec x)\}=\begin{cases} 1,\ 0<x<\frac{\pi}{2}\ \text{or}\ \frac{\pi}{2}<x<\pi\\ -1,\ \pi<x<\frac{3\pi}{2}\ \text{or}\ \frac{3\pi}{2}<x<2\pi \end{cases}

\displaystyle (xxxv)\ \frac{d}{dx}\{\cot^{-1}(\cot x)\}=1,\ (n-1)\pi<x<n\pi,\ n\in Z

 

\displaystyle 4.\ \text{Following are the fundamental rules for differentiation:}

\displaystyle (i)\ \frac{d}{dx}\{\text{Constant}\}=0

\displaystyle (ii)\ \frac{d}{dx}\{cf(x)\}=c\frac{d}{dx}\{f(x)\}

\displaystyle (iii)\ \frac{d}{dx}\{f(x)\pm g(x)\}=\frac{d}{dx}\{f(x)\}\pm\frac{d}{dx}\{g(x)\}

\displaystyle (iv)\ \frac{d}{dx}\{f(x)g(x)\}=f(x)\frac{d}{dx}\{g(x)\}+g(x)\frac{d}{dx}\{f(x)\}

\displaystyle (v)\ \frac{d}{dx}\left\{\frac{f(x)}{g(x)}\right\}=\frac{g(x)\frac{d}{dx}\{f(x)\}-f(x)\frac{d}{dx}\{g(x)\}}{\{g(x)\}^{2}}

\displaystyle (vi)\ \frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}

\displaystyle (vii)\ \frac{d}{dx}\left[\{f(x)\}^{g(x)}\right]=\{f(x)\}^{g(x)}\left\{\frac{g(x)}{f(x)}\frac{d}{dx}\{f(x)\}+\log f(x)\cdot\frac{d}{dx}\{g(x)\}\right\}

\displaystyle (viii)\ \text{If }x=\phi(t)\ \text{and}\ y=\psi(t),\ \text{then }\frac{dy}{dx}=\frac{dy/dt}{dx/dt}

\displaystyle (ix)\ \text{If }u\ \text{and}\ v\ \text{are functions of }x,\ \text{then }\frac{du}{dv}=\frac{du/dx}{dv/dx}

 

\displaystyle 5.\ \text{If }f(x),\ g(x),\ u(x)\ \text{and}\ v(x)\ \text{are function of }x\ \text{and }\Delta\text{ is a determinant given by}

\displaystyle \Delta(x)=\begin{vmatrix} f(x)&g(x)\\ u(x)&v(x) \end{vmatrix}.\ \text{Then,}

\displaystyle \frac{d}{dx}\{\Delta(x)\}=\begin{vmatrix} f'(x)&g'(x)\\ u(x)&v(x) \end{vmatrix}+\begin{vmatrix} f(x)&g(x)\\ u'(x)&v'(x) \end{vmatrix}

\displaystyle \text{Also,}\quad \frac{d}{dx}\{\Delta(x)\}=\begin{vmatrix} f'(x)&g(x)\\ u'(x)&v(x) \end{vmatrix}+\begin{vmatrix} f(x)&g'(x)\\ u(x)&v'(x) \end{vmatrix}

\displaystyle \text{Similar results hold for the differentiation of determinants of higher order.}

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