Integral using natural logarithm

Question

Can someone please tell the reason why you can't integrate $\int\frac{1}{e^x}\,\mathrm{d}x$ using the result that
$\int\frac{f'(x)}{f(x)}\,\mathrm{d}x$ = $\ln(f(x))+c$.
I alredy know that it gives the wrong integral, 1 which should have been actually $-e^{-x}$. But I can't see why the former method doesn't work?

Answer 1

You can actually do that (if you really want), you'd have to take $$f(x) = \exp(-\exp(-x)).$$

Answer 2

Because $\frac{df}{dx}= e^x$ not $= 1$