Find the derivative of the following function:
![\[ f(x) = e^{e^{e^x}}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-0254e6f48b804ede08c013d31c8e50a4_l3.png "Rendered by QuickLaTeX.com")
Using the chain rule and the formula for the derivative of an exponential we compute,
![\[ f'(x) = \left( e^{e^x} \right)' \left( e^{e^{e^x}} \right). \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-830e3de393fc21908219baeda4430110_l3.png "Rendered by QuickLaTeX.com")
From the previous exercise (Section 6.17, Exercise #11) (or by applying the chain rule and the formula for the derivative of the exponential again) we know
![\[ \left( e^{e^x} \right)' = e^x e^{e^x}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-1ff0218fe4ccf6778d24de69ffd36d78_l3.png "Rendered by QuickLaTeX.com")
Therefore,
![\[ f'(x) = e^x e^{e^x} e^{e^{e^x}}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-c662d23ff12c97c45d256e5ee6e038fa_l3.png "Rendered by QuickLaTeX.com")