Double Math

# Differentiation formulas with Proof.

## 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}