Question

f(x) =lin(x+linx)

Answer 1

$Given,\\f(x)=ln(x+lnx)$

and we know that $\dfrac{d}{dx}lnx=\dfrac{1}{x}$

So,

$f'(x)=\dfrac{d}{dx}ln(x+lnx)\\\ \\f'(x)=\dfrac{1}{(x+lnx)}\dfrac{d}{dx}[x+lnx]=\dfrac{1}{(1+lnx)}\cdot[1+\dfrac{1}{x}]=\dfrac{x+1}{x(1+lnx)}$

Hence, $\boxed{f'(x)=\dfrac{x+1}{x\cdot(1+lnx)}}$