## Calculus Concepts: An Informal Approach to the Mathematics of Change 5th Edition

inside: $g(x)=4 x^2$ outside: $f(g)=e^{g}$ derivative: $f^{\prime}(x)=8xe^{4x^2}$
Given$$f(x)=e^{4 x^2}$$ Use the chain rule to take the derivative $$\frac{d f(g(x))}{d x}=f^{\prime}(g(x)) g^{\prime}(x)$$ Here $g(x)=4x^2$ and $f(g)=e^g,$ then \begin{align*} f^{\prime}(x) &=\left(e^{g(x)}\right)^{\prime} \\ &=e^{g(x)} g^{\prime}(x) \\ &=8xe^{4x^2} \end{align*}