**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)}f(x)=lnxExamplederivative of lnxand graph

**differentiate w.r.t x**

**using ln property**

f(x)=\ln x^2Examplederivative of ln x power 2and graph

**differentiate w.r.t x**

**This is required derivative of ln of x square.**

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

**differentiate w.r.t x**