The derivative of a function is one of the basic and important concepts used in calculus and the process of finding a derivative is known as differentiation.
Derivative of a function
The derivative of a function (f), denoted by `f′ ` , is the function whose domain consists of those values of x such that the following limit exists:
`f'(x) =\lim_{h \to 0} \frac{f(x+h-f(x))}{h}`
Rules of Differentiation
Sum Rule:
If `y=u+v` , then `\frac{dy}{dx}=\frac{du}{dx}+\frac{dv}{dx}`
Difference Rule:
If $y=u-v$ , then $\frac{dy}{dx}= \frac{du}{dx}-\frac{dv}{dx}$
If $y=uv$ , then $\frac{dy}{dx}= u\frac{dv}{dx}+v\frac{du}{dx}$
Quotient Rule :
If $y=\frac{u}{v}$ , then $\frac{dy}{dx}=\frac{v\frac{du}{dx}+u\frac{dv}{dx}}{v^2}$
Chain Rule :
If $y= f(u)$ and $u=g(x)$ , $y=f(g(x))$ , then $\frac{dy}{dx}=\frac{d}{dx} (fg(x)) g'(x)$
Constant Rule:
If $y=ku$, then $\frac{dy}{dx}=k\frac{du}{dx}$ ,
where k =constant
Standard Derivative of Functions
1) $\frac{d}{dx}k =0 $ , where k= constant.
2) $\frac{d}{dx}x=1$
3) $\frac{d}{dx}x^n=n \cdot x^{n-1}$
4)$\frac{d}{dx}a^x= a^x \log{x}$
5) $\frac{d}{dx}e^x = e^x$
6) $\frac{d}{dx}(\frac{1}{x})=-\frac{1}{x^2}$
7) $\frac{d}{dx}\sqrt{x}= \frac{1}{2\sqrt{x}}$
8)$\frac{d}{dx} \log (x) = \frac{1}{x}$
9)$\frac{d}{dx} \sin x= \cos x$
10) $\frac{d}{dx} \cos x =- \sin x $
11) $\frac{d}{dx} \tan x = -\sec ^2 x $
12) $\frac{d}{dx} \cot x = -\csc ^2 x $
13) $\frac{d}{dx} \sec x = \sec x \times \tan x$
14) $\frac{d}{dx} \csc x= - \csc x \times \cot x$
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