Logarithmic Differentiation: Let [katex]f(x)=u^v[/katex] where both [katex]u[/katex] and [katex]v[/katex] are variables or function of [katex]x[/katex], the derivative of [katex]f(x)[/katex] can be obtained by taking natural logarithms of both of sides and the differentiating .
Example
[katex]f(x)=x^x[/katex]
[katex]lnf(x)=ln(x^x)[/katex]
now by usning property of ln
[katex]lnf(x)=x.lnx[/katex]
differentiate w.r.t x
[katex]\frac d{dx}\ln f(x)=\frac d{dx}x.\ln x[/katex]
property of ln and product rule
[katex]\frac1{f(x)}\frac d{dx}f(x)=\ln x\frac d{dx}x+x\frac d{dx}\ln x[/katex]
[katex]\frac{f'(x)}{f(x)}=\ln x(1)+x\frac1x[/katex]
[katex]\frac{f'(x)}{f(x)}=\ln x+1[/katex]
[katex]f'(x)=f(x).\left(\ln x+1\right)[/katex]
[katex]\boxed{f'(x)=x^x.\left(\ln x+1\right)}[/katex]
Example derivative of lnx and graph
[katex]f(x)=lnx[/katex]
differentiate w.r.t x
[katex]\frac d{dx}f(x)=\frac d{dx}\ln x[/katex]
using ln property
[katex]\boxed{f'(x)=\frac1x}[/katex]
Example derivative of ln x power 2 and graph
[katex]f(x)=\ln x^2[/katex]
differentiate w.r.t x
[katex]\frac d{dx}f(x)=\frac d{dx}\ln x^2[/katex]
[katex]f'(x)=\frac1{x^2}\frac d{dx}x^2[/katex]
[katex]f'(x)=\frac1{x^2}2x[/katex]
[katex]\boxed{f'(x)=\frac2x}[/katex]
This is required derivative of ln of x square.
Example derivative of ln(sinx) and graph
[katex]y=\ln(\sin x)[/katex]
differentiate w.r.t x
[katex]\frac d{dx}y=\frac d{dx}\ln(\sin x)[/katex]
[katex]\frac d{dx}y=\frac1{\sin x}.\frac d{dx}(\sin x)[/katex]
[katex]\frac d{dx}y=\frac1{\sin x}.\cos x[/katex]
[katex]\boxed{\frac d{dx}y=cotx}[/katex]