Question

F(x)=(lnx)²/2x give f '(x)=??

Answer 1

$f(x)=\frac{(lnx)^2}{2x};\ D_f:x\in\mathbb{R^+}\\\\use:\left[\frac{f(x)}{g(x)}\right]'=\frac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}\\\\f'(x)=\frac{[(lnx)^2]'\cdot2x-(lnx)^2\cdot(2x)'}{(2x)^2}=(*)\\\\\ [(lnx)^2]'=2lnx\cdot\frac{1}{x}=\frac{2lnx}{x}\\\\(2x)'=2\\\\(*)=\frac{\frac{2lnx}{x}\cdot2x-(lnx)^2\cdot2}{4x^2}=\frac{4lnx-2(lnx)^2}{4x^2}=\frac{[4lnx-2(lnx)^2]:2}{4x^2:2}=\frac{2lnx-(lnx)^2}{2x^2}$

Answer 2

$f(x)=\dfrac{(\ln x)^2}{2x}\\\\ f'(x)=\dfrac{2\ln x\cdot \dfrac{1}{x}\cdot2x-(\ln x)^2\cdot2}{(2x)^2}\\ f'(x)=\dfrac{2\ln x(2-\ln x)}{4x^2}\\ f'(x)=-\dfrac{\ln x(\ln x-2)}{2x^2}$