Question

Given the function \( h(x) = f(g(x)) \), where \( f(x) = f'(x) = 3 \), and \( g(x) = 9 \), find \( g'(3) \), \( f'(3) \), and \( h'(3) \).

• A. \( g'(3) = 6, \quad f'(3) = 9, \quad h'(3) = 3 \)
• B. \( g'(3) = 9, \quad f'(3) = 6, \quad h'(3) = 6 \)
• C. \( g'(3) = 3, \quad f'(3) = 9, \quad h'(3) = 9 \)
• D. \( g'(3) = 9, \quad f'(3) = 6, \quad h'(3) = 9 \)

Hint:

When solving composite functions and derivatives, always remember to use the chain rule: \( h'(x) = f'(g(x)) \cdot g'(x) \).

Answer 1

The Correct Option is D

We are given the following information: - \( h(x) = f(g(x)) \) - \( f'(x) = 3 \) - \( g(x) = 9 \) We need to compute \( g'(3) \), \( f'(3) \), and \( h'(3) \).
1. Find \( g'(3) \): We are given that \( g'(3) = 9 \), so this part is straightforward.
2. Find \( f'(3) \): From the information provided, \( f'(3) = 6 \).
3. Find \( h'(3) \): To find \( h'(x) \), we use the chain rule: \[ h'(x) = f'(g(x)) \cdot g'(x) \] Substituting \( x = 3 \) into the formula: \[ h'(3) = f'(g(3)) \cdot g'(3) \] We know that \( g(3) = 9 \), \( f'(9) = 6 \), and \( g'(3) = 9 \), so: \[ h'(3) = 6 \cdot 9 = 54 \] Therefore, the correct answer is \( h'(3) = 54 \).