We are given the function $y = e^{e^{x}}$. We need to find the derivative $\frac{dy}{dx}$ using the chain rule.
The chain rule states that if $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$. Here, let $u = e^{x}$, so $y = e^{u}$.
First, differentiate $e^{u}$ with respect to $u$, which is $e^{u}$. Then, differentiate $u = e^{x}$ with respect to $x$, which is $e^{x}$.
Multiplying these results together: $$ \frac{dy}{dx} = e^{e^{x}} \cdot \frac{d}{dx}(e^{x}) $$ $$ \frac{dy}{dx} = e^{e^{x}} \cdot e^{x} $$
Final Answer: $e^{x}e^{e^{x}}$
AI generated content. Review strictly for academic accuracy.