Let f and g be differentiable functions satisfying g'(a) = 2, g(a) = b and fog = I (identity function). Then, f'(b) is equal to
Answer & explanation
Correct answer: option 3
We have,
fog = I
⇒ fog(x) = I(x) for all x ∈ R
⇒ fog(x) = x for all x ∈ R
⇒ $\frac{d}{d x}(fog(x)) =1$ for all x ∈ R
⇒ f'(g(x)) g'(x) = 1 for all x ∈ R
⇒ f'(g(a)) g'(a) = 1
⇒ 2f'(b) = 1 [∵ g'(a) = 2 and g(a) = 6]
⇒ f'(b) = $\frac{1}{2}$