Intuition difference between derivatives of $\exp (x) $ and $\log (x)$ It is well known that the exponential function $\exp (x)$ has the derivative $$\frac{d}{dx} \exp (x) = \exp (x).$$ However, its inverse, the (natural) logarithm, $\log (x)$, in fact changes after derivation: $$\frac{d}{dx} \log (x) = \frac{1}{x}. $$
I understand why this is correct, but this is not intuitive to me. Why does $\exp (x) $ not change after derivation, while $\log (x) $ does and they are just mirrored at $y = x $?
EDIT: Thank you all for your answers, they really helped me understanding this!

If you know the chain rule, you can start from
$$
x = e^{\ln x}
$$
and take the derivative on both sides:
$$
1 = e^{\ln x}(\ln'x) = x\ln'x,
$$
such that $\ln'x = 1/x$. Alternatively,
$$
y = \ln(e^{y}),
$$
so that
$$
1 = e^{y}\ln'(e^{y}).
$$
Rename $x=e^y$, and you get the same result.
