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}

Spread the love
Azhar Ali

Azhar Ali

I graduated in Mathematics from the University of Sargodha, having master degree in Mathematics.

Leave a Reply

Your email address will not be published.

Mathematics is generally known as Math in US and Maths in the UK.

Contact Us

Copyright by Double Math. All Right Reserved 2019 to 2022