How to evaluate the derivative of $e^{e^x}$

Question

How do I evaluate the derivative of the following function?

$$ f(x) = e^{e^x} $$

Which kind of function is it? Exponential?

I tried to use the fact that $[e^x]' = e^x$ but I can't think about any strategy to solve $e^{e^x}$.

Answer 1

Letting $g(x) = e^x$, we have $f(x) = g \big( g(x) \big)$. Now apply the chain rule:

$$f'(x) = g' \big(g(x) \big) \cdot g'(x)$$

Answer 2

Another way of doing this calculation is the following: Let $y= e^{e^x}$. Notice that $\ln y = e^x$. Taking the derivatives on both sides of this identity we obtain $$\frac{y'}{y} = e^x$$ and therefore $$y'=ye^x = e^{e^x}e^x.$$

Answer 3

$y=e^{e^x}\Rightarrow \ln y= e^x\Rightarrow \frac{y'}{y}=e^x$

then $y'=ye^x\Rightarrow y'=e^{e^x}e^x=e^{e^x+x}$

Answer 4

The answers above certainly do the job but I think it may help to write explicitly what you are calculating

$$ \frac{d}{dx} e^{e^x} = \frac{d \; (e^{e^x})}{d \;e^x} \frac{d (e^x)}{dx}. $$

The first term

$$ \frac{d \; (e^{e^x})}{d \;(e^x)} = \frac{d \; (e^{f(x)})}{d \;f(x)} = e^{f(x)} = e^{e^x}, $$

and the second term is obvious so

$$ \frac{d}{dx} e^{e^x} = e^{e^x} e^x = e^{e^x + x}. $$