NTA Abhyas JEE Main2020MathematicsDifferentiationPractice

Consider a function f x = x x , ∀ x ∈ 1 , ∞ . If g x is the inverse function of f x , then the value of g ′ 4 is equal to

Options

Alog 2 ⁡ e
B1 2 log 2 e ⁡ e
C1 4 log 2 e ⁡ e
D1 2 log e ⁡ 2 e

Correct answer

C. 1 4 log 2 e ⁡ e

Step-by-step solution

f x = x x ⇒ log ⁡ f x = x log ⁡ x Differentiating with respect to x , we get, f ′ x = x x 1 + log x Now, by property g ′ f x = 1 f ′ x ∵ g x is the inverse of f x Putting x = 2 g ' f 2 = 1 f ' 2 ⇒ g ' 4 = 1 2 2 1 + log 2 = 1 4 log 2 e e

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