# Logarithmic Differentiation and its Examples with Solution

Logarithmic Differentiation: Let f(x)=u^v where both u and v are variables or function of x, the derivative of f(x) can be obtained by taking natural logarithms of both of sides and the differentiating .

Example

f(x)=x^x

lnf(x)=ln(x^x)

now by usning property of ln

lnf(x)=x.lnx

differentiate w.r.t x

\frac d{dx}\ln f(x)=\frac d{dx}x.\ln x

property of ln and product rule

\frac1{f(x)}\frac d{dx}f(x)=\ln x\frac d{dx}x+x\frac d{dx}\ln x

\frac{f'(x)}{f(x)}=\ln x(1)+x\frac1x

\frac{f'(x)}{f(x)}=\ln x+1

f'(x)=f(x).\left(\ln x+1\right)

\boxed{f'(x)=x^x.\left(\ln x+1\right)}
Example  derivative of lnx and graph
f(x)=lnx

differentiate w.r.t x

\frac d{dx}f(x)=\frac d{dx}\ln x

using ln property

\boxed{f'(x)=\frac1x}
Example  derivative of ln x power 2 and graph
f(x)=\ln x^2

differentiate w.r.t x

\frac d{dx}f(x)=\frac d{dx}\ln x^2

f'(x)=\frac1{x^2}\frac d{dx}x^2

f'(x)=\frac1{x^2}2x

\boxed{f'(x)=\frac2x}

This is required derivative of ln of x square.

Example derivative of ln(sinx) and graph
y=\ln(\sin x)

differentiate w.r.t x

\frac d{dx}y=\frac d{dx}\ln(\sin x)

\frac d{dx}y=\frac1{\sin x}.\frac d{dx}(\sin x)

\frac d{dx}y=\frac1{\sin x}.\cos x

\boxed{\frac d{dx}y=cotx} 