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Consider the function f x = t a n - 1 3 x - 2 3 + 2 x , ∀ x ≥ 0 . If g x is the inverse function of f x , then the value of g ′ π 4 is equal to

Correct answer

26

Step-by-step solution

f x = t a n - 1 x - 2 3 1 + x . 2 3 = t a n - 1 x - t a n - 1 2 3 ⇒ f ' x = 1 1 + x 2     . . .     1 Now, g ' f x = 1 f ' x   ∵   g f x = x Putting x = 5 , we get, g ' f 5 = 1 f ' 5 ⇒ g ' tan − 1 3 5 − 2 3 + 2 5 = 1 1 1 + 5 2 ⇒ g ' tan − 1 1 = 26 ⇒ g ' π 4 = 26

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