Answer
inside: $g(x)= \ln 6x$
outside: $f(g)=g^{2}$
derivative: $f^{\prime}(x)=\dfrac{2\ln 6 x}{ x}$
Work Step by Step
Given $$f(x)=(\ln 6 x)^{2}$$
Use the chain rule to take the derivative
$$
\frac{d f(g(x))}{d x}=f^{\prime}(g(x)) g^{\prime}(x)
$$
Here $g(x)=\ln 6 x$ and $f(g)=g^{2},$ then
\begin{align*}
f^{\prime}(x) &=\left(g^{2}(x)\right)^{\prime} \\ &=2 g (x) g^{\prime}(x) \\ &=2\left(\ln 6 x\right)\frac{6}{6x}\\
&=\frac{2\ln 6 x}{ x}
\end{align*}
