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The problem asks to evaluate \( h'(8) \), where \( h(x) = f(x) \cdot g(x) \), and it gives the following information:

- \( f(8) = 9 \)
- \( f'(8) = -1.5 \)
- \( g(8) = 4 \)
- \( g'(8) = 3 \)

Since \( h(x) = f(x) \cdot g(x) \), we need to apply the product rule to find \( h'(x) \). The product rule states:

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

Now we substitute the given values into the equation for \( h'(8) \):

\[
h'(8) = f'(8)g(8) + f(8)g'(8)
\]
\[
h'(8) = (-1.5)(4) + (9)(3)
\]
\[
h'(8) = -6 + 27
\]
\[
h'(8) = 21
\]

Thus, \( h'(8) = 21 \).

Would you like further details or have any questions?

Here are 5 related questions:
1. What is the product rule in differentiation?
2. How would you apply the quotient rule for division of functions?
3. What happens if one of the functions is a constant, how does the derivative change?
4. How do second derivatives work with product rule applications?
5. Can the chain rule be used in combination with the product rule?

**Tip:** When using the product rule, always carefully check that you apply the correct derivatives to each function before multiplying!

Evaluate h'(8) where h(x) = f(x) ⋅ g(x), given values for f(8), f'(8), g(8), and g'(8).

This problem evaluates h'(8) using the product rule, where h(x) = f(x) ⋅ g(x). Given f(8) = 9, f'(8) = -1.5, g(8) = 4, and g'(8) = 3, learn how to apply the product rule for differentiation to solve for h'(8) = 21. Suitable for students in Grades 11-12 or college-level calculus.

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