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!