Let  
h
(
x
)
=
f
(
x
)
g
(
x
)
 . If  
f
(
x
)
=
x
2
+
8
 ,  
g
(
−
3
)
=
8
 , and  
g
′
(
−
3
)
=
−
3
 . Find  
h
′
(
−
3
)
 .

We are asked to find \( h'(-3) \) where \( h(x) = f(x) g(x) \), and the following information is provided:

- \( f(x) = x^2 + 8 \)
- \( g(-3) = 8 \)
- \( g'(-3) = -3 \)

To solve for \( h'(x) \), we apply the **product rule** of differentiation, which states:
\[
h'(x) = f'(x) g(x) + f(x) g'(x)
\]

### Step 1: Differentiate \( f(x) \)
Since \( f(x) = x^2 + 8 \), we differentiate it with respect to \( x \):
\[
f'(x) = 2x
\]

### Step 2: Use the product rule
We can now substitute into the product rule formula:
\[
h'(-3) = f'(-3) g(-3) + f(-3) g'(-3)
\]

### Step 3: Calculate each term
- \( f'(-3) = 2(-3) = -6 \)
- \( g(-3) = 8 \) (given)
- \( f(-3) = (-3)^2 + 8 = 9 + 8 = 17 \)
- \( g'(-3) = -3 \) (given)

Now substitute these values into the product rule:
\[
h'(-3) = (-6)(8) + (17)(-3)
\]
\[
h'(-3) = -48 + (-51) = -99
\]

Thus, the value of \( h'(-3) \) is:
\[
h'(-3) = -99
\]

Would you like further details or have any questions about this? 

Here are 5 related questions you might find helpful:
1. How do you apply the product rule to more complicated functions?
2. Can the product rule be applied to more than two functions?
3. What if \( g(x) \) or \( f(x) \) is a trigonometric function, how does that affect the derivative?
4. How do you calculate second derivatives for product functions?
5. What is the chain rule and how does it relate to the product rule?

**Tip:** Remember to always check if you need to apply more than one differentiation rule (e.g., product and chain rules together).

Let h(x) = f(x)g(x). If f(x) = x^2 + 8, g(-3) = 8, and g'(-3) = -3, find h'(-3).

Learn how to apply the product rule to differentiate h(x) = f(x)g(x) where f(x) = x^2 + 8. Given g(-3) = 8 and g'(-3) = -3, find h'(-3) with step-by-step instructions. This problem is suitable for high school students in Grades 10-12 and involves concepts in differentiation.

Math Problem Statement

Let
h ( x )

f ( x ) g ( x ) . If
f ( x )

x 2 + 8 ,
g ( − 3 )

8 , and
g ′ ( − 3 )

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