Chapter 7 - Exponential Functions - 7.1 Derivative of f(x)=bx and the Number e - Exercises - Page 327: 40

\begin{align*}f'(x)&=e^x e^{e^x} . \end{align*}

Recall that $(e^x)'=e^x$ Since we have $$ f(x)= e^{e^x}$$ then the derivative $ f'(x)$, using the chain rule, is given by \begin{align*}f'(x)&=e^{e^x} (e^x)'\\ &=e^x e^{e^x} . \end{align*}