Question

If \( f \) and \( g \) are differentiable functions satisfying \( g'(a) = 2 \), \( g(a) = b \), and \( f \circ g = 1 \), where 1 is an identity function, then \( f'(b) \) is equal to

• A. \( \frac{1}{2} \)
• B. \( \frac{3}{2} \)
• C. \( \frac{2}{3} \)
• D. 2

Hint:

When differentiating composite functions, apply the chain rule and solve for the derivative of the outer function.

Answer 1

The Correct Option is A

Step 1: Use the chain rule for the composite function.
We are given that \( f \circ g = 1 \), meaning \( f(g(x)) = 1 \). Differentiating both sides with respect to \( x \) gives: \[ f'(g(x)) \cdot g'(x) = 0 \] Since \( g'(a) = 2 \), we have: \[ f'(g(a)) \cdot 2 = 0 \] Thus, \( f'(b) = \frac{1}{2} \).
Step 2: Conclusion.
Therefore, the value of \( f'(b) \) is \( \frac{1}{2} \).