Answer
inside: $g(x)=0.08x$
outside: $f(g)=1+58e^{g}$
derivative: $f^{\prime}(x)=4.64 e^{0.08x}$
Work Step by Step
Given$$
f(x)=1+58 e^{0.08 x}
$$
Use the chain rule to take the derivative
$$
\frac{d f(g(x))}{d x}=f^{\prime}(g(x)) g^{\prime}(x)
$$
Here $g(x)=0.08x$ and $f(g)=1+58e^g,$ then
\begin{align*}
f^{\prime}(x) &=\left(1+58e^{g(x)}\right)^{\prime} \\
&=58e^{g(x)} g^{\prime}(x) \\
&=58(0.08)e^{0.08x}\\
&=4.64 e^{0.08x}
\end{align*}