Proof for $\int \frac{f'(x)}{f(x)} \, \mathrm{d}x= \ln f(x) + c$

Question

I know that the proof for:

$$\int \frac{f'(x)}{f(x)}\ dx = \ln(f(x)) + C$$

has something to do with the chain rule. However I'm yet to find a clear explanation for this among existing threads. Will be glad for help.

Answer 1

Hint: What is the derivative of $\ln(f(x))$, according to the chain rule?

Answer 2

Alternatively, use the substitution $u = f(x)$ and $\frac{\mathrm{d}u}{\mathrm{d}x} = f'(x)$, so your integral is

$$\int \frac{\mathrm{d}u}{u} = \ln u+ \mathcal{C} = \ln f(x) + \mathcal{C}$$