Use the chart to find h, prime, of, 1h 
′
 (1) in simplest form.
xx	f, of, xf(x)	f, prime, of, xf 
′
 (x)	g, of, xg(x)	g, prime, of, xg 
′
 (x)
11	minus, 10−10	33	minus, 7−7	77
minus, 7−7	minus, 2−2	minus, 4−4	minus, 5−5	1111
h, of, x, equals, f, of, g, of, x
h(x)=
f(g(x))


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.

Use the chart to find h'(1) in simplest form, where h(x) = f(g(x)).

Learn how to apply the chain rule to find h'(1) where h(x) = f(g(x)) using a given chart. This calculus problem demonstrates the steps to differentiate composite functions, using the derivative formula h'(x) = f'(g(x)) * g'(x). Suitable for Grades 11-12 and introductory college calculus.

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