Differentiation formulas with Proof.

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Derivative of product rule or differentiation of product rule

let $f(x)=u.v$

where $u$ and $v$ are function of $x$

$\frac d{dx}f\left(x\right)=\frac d{dx}\left[\left(u\right).\left(v\right)\right]$

$\boxed{\frac d{dx}f\left(x\right)=\left(u\right).\frac d{dx}\left(v\right)+\left(v\right).\frac d{dx}\left(u\right)}$

Derivative of quotient rule or differentiation of quotient rule

$f\left(x\right)=\frac uv$

where $u$ and $v$ are function of $x$

$\frac d{dx}f\left(x\right)=\frac d{dx}\frac uv$

$\boxed{\frac d{dx}f\left(x\right)=\frac{\left(v\right).{\displaystyle\frac d{dx}}\left(u\right)-\left(u\right).{\displaystyle\frac d{dx}}\left(v\right)}{\left(v\right)^2}}$

Derivative of power rule or differentiation of power rule

$f\left(x\right)=x^n$

$\frac d{dx}f\left(x\right)=\frac d{dx}x^n$

$\boxed{\frac d{dx}f\left(x\right)=nx^{n-1}}$

Derivative of chain rule or differentiation of chain rule

Let $y=f(u)$ and $u=g(x)$

$\boxed{\frac{dy}{dx}=\frac{dy}{du}.\frac{du}{dx}}$

Derivative of trigonometric functions or Differentiation of trigonometric functions

$\boxed{\frac d{dx}\left(\sin x\right)=\cos x}$

$\boxed{\frac d{dx}\left(\cos x\right)=-sin x}$

$\boxed{\frac d{dx}\left(\tan x\right)=sec^2 x}$

$\boxed{\frac d{dx}\left(\cot x\right)=-cosec^2 x}$

$\boxed{\frac d{dx}\left(\cosec x\right)=-cosec x.cot x}$

$\boxed{\frac d{dx}\left(\sec x\right)=sec x.tan x}$

Derivative of inverse trigonometric functions ,Differentiation of inverse trigonometric functions

$\boxed{\frac d{dx}(Sin^{-1}x)=\frac1{\sqrt{1-x^2}}}$

$\boxed{\frac d{dx}(Cos^{-1}x)=\frac{-1}{\sqrt{1-x^2}}}$

$\boxed{\frac d{dx}(Tan^{-1}x)=\frac{1}{{1+x^2}}}$

$\boxed{\frac d{dx}(Cot^{-1}x)=\frac{-1}{{1+x^2}}}$

$\boxed{\frac d{dx}(sec^{-1}x)=\frac1{\left|x\right|\sqrt{1-x^2}}}$

$\boxed{\frac d{dx}(Cosec^{-1}x)=\frac{-1}{\left|x\right|\sqrt{1-x^2}}}$

Derivative of sum and difference rule, Differentiation of sum and difference rule

sum rule of Derivative

let $y=u(x)+v(x)$

where $u$ and $v$ are function of $x$

$\boxed{\frac d{dx}\left(y\right)=\frac d{dx}\left[u\left(x\right)+v\left(x\right)\right]}$

difference rule of Derivative

let $y=u(x)-v(x)$

where $u$ and $v$ are function of $x$

$\boxed{\frac d{dx}\left(y\right)=\frac d{dx}\left[u\left(x\right)-v\left(x\right)\right]}$

Derivative of hyperbolic functions

$\boxed{Sinhx=\frac{e^x-e^{-x}}2}$

$\boxed{Coshx=\frac{e^x+e^{-x}}2}$

$\boxed{Tanhx=\frac{Sinhx}{Coshx}=\frac{e^x+e^{-x}}{e^x-e^{-x}}}$

$\boxed{Cosechx=\frac1{Sinhx}=\frac2{e^x-e^{-x}}}$

$\boxed{Sechx=\frac1{Coshx}=\frac2{e^x+e^{-x}}}$

$\boxed{Cothx=\frac{Coshx}{Sinhx}=\frac{e^x+e^{-x}}{e^x-e^{-x}}}$

Logarithmic differentiation

$\boxed{\frac d{dx}\ln x=\frac1x}$

Derivative of constant is zero

$\boxed{\frac{dy}{dx}=0}$

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