NTA Abhyas JEE Main2020MathematicsDifferentiationPractice

Let ϕ ( x ) be the inverse of the function f ( x ) and f ′ x = 1 1 + x 5 , then d d x ϕ x is equal to

Options

A1 1 + ϕ x 5
B1 1 + f x 5
C1 + ϕ x 5
D1 + f ( x )

Correct answer

C. 1 + ϕ x 5

Step-by-step solution

We have, ϕ x = f - 1 ( x ) ⇒ x = f [ ϕ x ] On differentiating both sides w.r.t. x , we get 1 = f ' ϕ x . ϕ ' ( x ) ⇒ ϕ ' x = 1 f ' ϕ x ...(i) Since, f ′ x = 1 1 + x 5 (given) f ' ϕ x = 1 1 + ϕ x 5 From Equation (i), ϕ ' x = 1 f ' [ ϕ x ] = 1 + ϕ x 5

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