Let's analyze the problem from the image.

You are asked to compute two derivatives using the given table for the functions $f(x)$, $g(x)$, $f'(x)$, and $g'(x)$:

1. $h'(4)$ where $h(x) = f(g(x))$
2. $p'(3)$ where $p(x) = g(f(x))$

Part a: $h'(4) = f'(g(4)) \cdot g'(4)$

By the chain rule, for $h(x) = f(g(x))$: $h'(x) = f'(g(x)) \cdot g'(x)$ Substitute $x = 4$: $h'(4) = f'(g(4)) \cdot g'(4)$

From the table:

• $g(4) = 5$
• $g'(4) = 2$
• $f'(5) = 3$

Thus: $h'(4) = f'(5) \cdot g'(4) = 3 \cdot 2 = 6$

Part b: $p'(3) = g'(f(3)) \cdot f'(3)$

For $p(x) = g(f(x))$, again using the chain rule: $p'(x) = g'(f(x)) \cdot f'(x)$ Substitute $x = 3$: $p'(3) = g'(f(3)) \cdot f'(3)$

From the table:

• $f(3) = 2$
• $f'(3) = 3$
• $g'(2) = 4$

Thus: $p'(3) = g'(2) \cdot f'(3) = 4 \cdot 3 = 12$

Final Answers:

• $h'(4) = 6$
• $p'(3) = 12$

Let me know if you need further details or explanations.

5 Related Questions:

1. How do you apply the chain rule to composite functions in general?
2. What would $h'(2)$ be based on the given table?
3. Can you calculate $p'(4)$ with the same method?
4. How is the chain rule derived from the definition of a derivative?
5. What is the product rule, and how is it different from the chain rule?

Tip:

When dealing with derivatives of composite functions, always remember to apply the chain rule carefully—differentiate the outer function, then multiply by the derivative of the inner function.