Question

if f(x) is a function whose derivative is f'(x)=1/x, find the derivative of the function y=xf(x)-x

Answer 1

$\bf \textit{let's make }f(x)=z\qquad thus\quad z'(x)=\cfrac{1}{x}\\\\ \textit{for the sake of disambiguiting it some} \\\\ -----------------------------\\\\ y=xz-x\impliedby \textit{using the product rule} \\\\\\ \cfrac{dy}{dx}=[1\cdot z+x\cdot \frac{dz}{dx}]-1\implies \cfrac{dy}{dx}=z+x\cdot \cfrac{1}{x}-1\implies \cfrac{dy}{dx}=z$