Differentiating powers of e where the power is an exponential function like $e^{a^x}$ when $a$ is a constant

Question

Using the fact that $e = \lim_{h \to \infty} (1+h)^{1/h}$, you can answer $\frac{d}{dx} e^{p(x)}$ where $p(x)$ is a polynomial using the variable x. However, I have troubles finding the derivative of $e^{p(x)}$ where “something” is a exponential function. Is there a way/formula (or multiple formulae) finding the derivative of such functions? I’ll be glad if you include the proof of such ways.

Answer 1

A useful way for this kind of problem is logarithmic differentiation.

$$y=e^{a^{f(x)}} \implies \log(y)=a^{f(x)}=e^{f(x)\log(a)}\implies\frac {y'}y=e^{f(x)\log(a)}\log(a) f'(x)$$