Find the derivative of $LaTeX: \displaystyle f(x) = e^{e^{e^{x}}}$

Taking the derivative with the chain rule gives $LaTeX: \displaystyle f'(x) = \frac{df}{du}\frac{du}{dv}\frac{dv}{dx}$ . $LaTeX: \displaystyle f'(x) = (e^{u})(e^{v})(e^{x})$ . Substituting back in $LaTeX: \displaystyle u$ and $LaTeX: \displaystyle v$ gives $LaTeX: \displaystyle f'(x) = e^{v} e^{x} e^{e^{v}} = e^{x} e^{e^{x}} e^{e^{e^{x}}}$ .