Question

Let \( f \) and \( g \) be differentiable functions such that \( f(3)=5, g(3)=7, f'(3)=13, g'(3)=6, f'(7)=2 \) and \( g'(7)=0 \). If \( h(x) = (f \circ g)(x) \), then \( h'(3) \) is

• A. \( 14 \)
• B. \( 12 \)
• C. \( 16 \)
• D. \( 0 \)
• E. \( 10 \)

Hint:

Always evaluate inner function first in chain rule.

Answer 1

The Correct Option is B

Concept: Chain rule: \[ (f \circ g)'(x) = f'(g(x)) \cdot g'(x) \]

Step 1: Identify function composition.
\[ h(x) = f(g(x)) \]

Step 2: Apply chain rule.
\[ h'(x) = f'(g(x)) \cdot g'(x) \]

Step 3: Evaluate inner function at \( x=3 \).
\[ g(3) = 7 \]

Step 4: Substitute into derivative.
\[ h'(3) = f'(7) \cdot g'(3) \]

Step 5: Use given values.
\[ = 2 \cdot 6 = 12 \]