Answer
inside: $g(x)=1+9.8 e^{-1.2x}$
outside: $f(g)=37.5(g(x) )^{-1}$
derivative: $f^{\prime}(x)=441e^{-1.2x}(1+9.8 e^{-1.2 x} )^{-2}$
Work Step by Step
Given$$
f(x)=\frac{37.5}{1+9.8 e^{-1.2 x}}
$$
Rewriting $f(x)$ as $$ f(x)= 37.5(1+9.8 e^{-1.2x})^{-1}$$
Use the chain rule to take the derivative
$$
\frac{d f(g(x))}{d x}=f^{\prime}(g(x)) g^{\prime}(x)
$$
Here $g(x)=1+9.8 e^{-1.2x}$ and $f(g)= 37.5(g(x) )^{-1},$ then
\begin{align*}
f^{\prime}(x) &=\left(37.5(g(x) )^{-1}\right)^{\prime} \\
&=-37.5(g(x) )^{-2}g^{\prime}(x) \\
&=-37.5(1+9.8 e^{-1.2 x} )^{-2}((-1.2)9.8e^{-1.2 x}) \\
&=441e^{-1.2x}(1+9.8 e^{-1.2 x} )^{-2}
\end{align*}
