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

2 $\frac{2}{3}$ $\frac{1}{2}$ none of these

$\frac{1}{2}$

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) = $\frac{1}{2}$