Question

If g(x) is the inverse function of f(x) and f'(x) = `1/(1 + x^4)`, then g'(x) is ______.

Options

• 1 + [g(x)]⁴

• 1 – [g(x)]⁴

• 1 + [f(x)]⁴

• `1/(1 + [g(x)]^4`

Answer 1

If g(x) is the inverse function of f(x) and f'(x) = `1/(1 + x^4)`, then g'(x) is 1 + [g(x)]⁴.

Explanation:

Given, g(x) = f^–1(x)

f(g(x)) = x

On differentiating both sides w.r.t. ‘x’, we get

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

∴ `1/(1 + (g(x))^4) g^'(x)` = 1  ...`[∵ f^'(x) = 1/(1 + x^4)]`

⇒ g'(x) = 1 + [g(x)]⁴