1. **State the problem:** We are given a function $h$ and its derivative $h'$ at certain points, and a function $g$ such that $h(g(x)) = x$ for all $x$. We need to find $g'(7)$. 2. **Recall the formula:** Since $h(g(x)) = x$, $g$ is the inverse function of $h$. The derivative of the inverse function is given by: $$g'(x) = \frac{1}{h'(g(x))}$$ 3. **Find $g(7)$:** Since $h(g(7)) = 7$, we look for $x$ such that $h(x) = 7$. From the table: - When $x=3$, $h(3) = 7$ So, $g(7) = 3$. 4. **Find $h'(g(7))$:** We need $h'(3)$. From the table: - $h'(3) = 5$ 5. **Calculate $g'(7)$:** $$g'(7) = \frac{1}{h'(3)} = \frac{1}{5}$$ 6. **Answer:** The value of $g'(7)$ is $\frac{1}{5}$, which corresponds to option C.