To find $h'(2)$ using the given chart and the fact that $h(x) = f(g(x))$, we apply the chain rule for derivatives:

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

Step-by-step solution:

1. Evaluate $g(2)$: From the chart, $g(2) = -5$.

2. Evaluate $g'(2)$: From the chart, $g'(2) = -4$.

3. Evaluate $f'(g(2))$: Since $g(2) = -5$, we find $f'(-5)$ from the chart. From the chart, $f'(-5) = 7$.

4. Combine results: Using the chain rule formula: $h'(2) = f'(g(2)) \cdot g'(2)$ Substituting the values: $h'(2) = f'(-5) \cdot g'(2) = 7 \cdot (-4) = -28$

Final Answer:

$h'(2) = -28$

Would you like a more detailed explanation or help with a related question?

Related Questions:

1. What is the general application of the chain rule in composite functions?
2. How would the process change if $h(x) = g(f(x))$?
3. Can you explain the significance of the derivative of $g(x)$ in this problem?
4. What happens to $h'(x)$ if $g'(x)$ is zero at some point?
5. How can this method be extended to higher-order derivatives of $h(x)$?

Tip:

When dealing with composite functions, always identify the inner and outer functions first to correctly apply the chain rule.