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
differentiate w.r.t x
\frac d{dx}f(x)=\frac d{dx}\ln xusing ln property
\boxed{f'(x)=\frac1x}Example derivative of ln x power 2 and graph
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
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}