If f(g(x))= x, where f and g are inverses of each other

Solution

We are told that f and g are inverse functions , i.e., f(g(x)) = x for all x in the domain of g. Differentiate both sides with respect to x using the chain rule : ⇒ d/dx [f(g(x))] = d/dx [x] ⇒ f ′(g(x)) · g ′(x) = 1 Now solve for g ′(x): ⇒ g ′(x) = 1 / f ′(g(x))  … (i) We are given: f ′(x) = 1 / (1 + x²) Substitute x with g(x): ⇒ f ′(g(x)) = 1 / (1 + [g(x)]²) Now substitute into equation (i): ⇒ g ′(x) = 1 / [1 / (1 + [g(x)]²)] ⇒ g ′(x) = 1 + [g(x)]²