The problem you've provided involves finding the derivative $h'(-5)$ for the function $h(x) = f(g(x))$ using the given values:

• $f'(9) = 4$
• $g'(-5) = 2$
• $g(-5) = 9$
• $f(9) = 12$

We are tasked with using these values to find $h'(-5)$.

Step-by-step solution:

Given $h(x) = f(g(x))$, we can apply the chain rule to differentiate $h(x)$:

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

Now, let's substitute $x = -5$:

$h'(-5) = f'(g(-5)) \cdot g'(-5)$

From the given information:

• $g(-5) = 9$, so $h'(-5) = f'(9) \cdot g'(-5)$
• $f'(9) = 4$
• $g'(-5) = 2$

Thus, we can now substitute these values:

$h'(-5) = 4 \cdot 2 = 8$

Final Answer:

$h'(-5) = 8$

Would you like more details or have any questions? Here are five related questions to expand your understanding:

1. How does the chain rule apply when differentiating composite functions?
2. What are the general steps for finding the derivative of a function like $h(x) = f(g(x))$?
3. Why is knowing $g(-5)$ and $f'(9)$ important for this problem?
4. How do derivatives of composite functions change when the inner function is more complex?
5. How would you approach this if $f'(9)$ or $g'(-5)$ were different values?

Tip:

Whenever you're working with composite functions, always identify the inner and outer functions clearly and apply the chain rule step-by-step.