Maths. Section B1Continuity and Differentiability

Let g(x) be the inverse of the function f(x) and f'(x) = 1/(1+x^3). T…

Let g(x) be the inverse of the function f(x) and f'(x) = $\frac{1}{1+x^3}$. Then g'(x) is :

Answer & explanation

Correct answer: option 3

Since g(x) is the inverse of f(x)

∴ (gof)(x) = x

⇒ g(f(x)) = x

⇒ g'(f(x)) f'(x) = 1

⇒ $g'(f(x))=\frac{1}{f'(x)}=1+x^3$

⇒ g'(y) = 1 + (g(y))²

[∵ y = f(x) ⇒ x = f^–1(y) = g(y)]

⇒ g'(x) = 1 + (g(x))³. (Replacing y by x)

Hence (3) is correct answer.