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We are given the function \( h(x) = f(g(x)) \), and the goal is to find \( h'(2) \).

To find \( h'(x) \), we need to use the chain rule since \( h(x) \) is a composition of two functions, \( f(x) \) and \( g(x) \). The chain rule for derivatives tells us:

\[
h'(x) = f'(g(x)) \cdot g'(x)
\]

Now, we will apply this to \( h'(2) \):

1. From the table, find \( g(2) \). According to the table, \( g(2) = 3 \).
2. Now, find \( f'(g(2)) = f'(3) \). According to the table, \( f'(3) = 5 \).
3. Finally, find \( g'(2) \). From the table, \( g'(2) = 5 \).

Now apply the chain rule:

\[
h'(2) = f'(g(2)) \cdot g'(2) = f'(3) \cdot g'(2) = 5 \cdot 5 = 25
\]

Thus, the value of \( h'(2) \) is \( 25 \).

Do you need further clarification or have any other questions?

Here are 5 related questions to deepen your understanding:
1. How would you apply the chain rule if \( h(x) = f(g(x) + x) \)?
2. What happens to the derivative if \( g(x) \) is a constant function?
3. Can you apply the product rule in combination with the chain rule in this scenario?
4. How would you find \( h''(2) \) for the same functions?
5. What is the meaning of the chain rule in terms of rate of change for composed functions?

**Tip**: When using the chain rule, always start by identifying the outer function and the inner function to differentiate correctly!

Suppose h(x) = f(g(x)). Given the table of values below, determine h'(2).

Learn how to apply the chain rule to find the derivative h'(2) where h(x) = f(g(x)). Using the given table of values, we show the step-by-step process to solve for h'(2), demonstrating core concepts in calculus. Suitable for students in Grades 11-12 or those taking introductory calculus.

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