Recall that $f(g(x)) = x$ is the definition of inverse functions. In other words, you are being told that $f(x)$ and $g(x)$ are inverses.

Inverse functions have reciprocal slopes at corresponding $(x, y)→(y, x)$ values.
For example, if $f(x)$ has coordinate point $(2, 3) \text{ and }f'(2)=\frac{4}{5},\text{ then }g'(3)=\frac{5}{4}$