1. The problem is to find the derivative of the inverse function $g(x)$ of $f(x) = 5x + 2$ at $x=12$. 2. Recall that if $g$ is the inverse of $f$, then $g(f(x))=x$ and also $g'(y) = \frac{1}{f'(g(y))}$. 3. First, find the derivative of $f(x)$: $$ f'(x) = 5 $$ This derivative is constant because $f(x)$ is a linear function. 4. We want to find $g'(12)$. Using the formula: $$ g'(12) = \frac{1}{f'(g(12))} $$ 5. Next, find $g(12)$ by solving $f(x) = 12$: $$ 5x + 2 = 12 $$ $$ 5x = 10 $$ $$ x = 2 $$ So, $g(12) = 2$. 6. Evaluate $f'(g(12))$: $$ f'(2) = 5 $$ 7. Substitute back to find $g'(12)$: $$ g'(12) = \frac{1}{5} $$ Final answer: $\boxed{\frac{1}{5}}$ which corresponds to option D.