To find the value of $h'(3)$, where $h(x) = f(g(x))$, we need to use the chain rule for differentiation. The chain rule states that if $h(x) = f(g(x))$, then:

$h'(x) = f'(g(x)) \cdot g'(x)$

To determine $h'(3)$, we substitute $x = 3$:

$h'(3) = f'(g(3)) \cdot g'(3)$

From the table, we have the following values:

• $g(3) = 5$
• $g'(3) = 7$

Now, we need $f'(g(3)) = f'(5)$. From the table, $f'(5) = 4$.

Now, substitute these values into the expression for $h'(3)$:

$h'(3) = f'(5) \cdot g'(3) = 4 \cdot 7 = 28$

Therefore, the value of $h'(3)$ is:

$h'(3) = 28$

Would you like additional details or have questions?

Here are some related questions to explore further:

1. How does the chain rule apply to other compositions, like $k(x) = g(f(x))$?
2. What changes if the functions $f$ and $g$ were not differentiable?
3. How would you interpret $h(x) = f(g(x))$ graphically?
4. What if $g(x)$ were a constant function? How would $h'(x)$ behave?
5. Can the chain rule be extended to compositions involving more than two functions?

Tip: For compositions, always differentiate the outer function evaluated at the inner function times the derivative of the inner function.