To solve for $h'(1)$ using the given chart and the function $h(x) = f(g(x))$, we need to apply the chain rule. The chain rule for the derivative of a composition of functions states that:

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

We are asked to find $h'(1)$, so we will substitute $x = 1$ into the equation and use the values from the chart.

Step 1: Find $g(1)$

From the chart, we see that: $g(1) = 7$

Step 2: Find $f'(g(1)) = f'(7)$

From the chart, we know: $f'(7) = -5$

Step 3: Find $g'(1)$

From the chart, we see: $g'(1) = -2$

Step 4: Apply the chain rule

Now, we can plug in the values into the chain rule formula: $h'(1) = f'(g(1)) \cdot g'(1) = f'(7) \cdot g'(1) = (-5) \cdot (-2) = 10$

Thus, $h'(1) = 10$.

Would you like further details or clarification on any step?

Here are 5 related questions for you:

1. How would the result change if $g(1)$ had a different value?
2. Can you explain how the chain rule works in more complex compositions of functions?
3. What if $f(x)$ had a second derivative—how would you find $h''(x)$?
4. Could the function $g(x)$ affect the concavity of $h(x)$?
5. How would you interpret $h'(1)$ in a real-world context if this were a physics problem?

Tip: Always verify both $f'(x)$ and $g'(x)$ before applying the chain rule to avoid small mistakes.