Explain Why $\int\frac{f'(x)}{f(x)}\,\mathrm dx = \ln|f(x)|+C$ Explain why
$$\int\frac{f'(x)}{f(x)}\,\mathrm dx = \ln|f(x)|+C.$$

Thanks!

Differentiate RHS: you get $\frac{d}{dx}(\log |f(x)|+C)=\frac{d}{dx}(\log |f(x)|)=\frac{1}{|f(x)|}\cdot\frac{d}{dx}|f(x)|=\frac{f'(x)}{|f(x)|}\cdot Sgn(f(x))=\frac{f'(x)}{f(x)}$, where $Sgn(f)=1$ where $f\geq 0$, $Sgn(f)=-1$ where $f<0$
