Let ƒ and g be differentiable functions on R such that fog is the identity function. If for some a, b ∈ R, g'(a) = 5 and g(a) = b, then ƒ'(b) is equal to :

Solution

Maths sir · 4 steps

Differentiate f(g(x)) = x with the chain rule, then put x = a; everything else falls out.

1. Write the condition. f ° g is the identity, so f(g(x)) = x for every real x.
2. Differentiate both sides. By the chain rule, the derivative of f(g(x)) is f'(g(x)) × g'(x). The derivative of x is 1.

   f'(g(x)) g'(x) = 1

3. Put x = a. Since g(a) = b and g'(a) = 5, this becomes f'(b) × 5 = 1.
4. So. f'(b) = 1/5, which is option (d).

The idea to keep. If f(g(x)) = x, then f' at the point g(a) is the reciprocal of g'(a). Always evaluate f' at g(a), never at a.

Answer from the NTA final answer key. Solution by PrepWiser.