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

Correct answer: C. 1 + [ϕ(x)] ^ 5 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

JEE Main

Exam: JEE Main
Subject: Mathematics
Chapter: Differentiation

JEE Main · Mathematics · Differentiation

Let ϕ ( x ) be the inverse of the function f ( x ) and f^ ′ (x) = 1...

Let

ϕ ( x )

be the inverse of the function

f ( x )

and

f^{′} \left(x\right) =

\dfrac{1}{1 + x^{5}},then \dfrac{d}{dx}

ϕ\left(x\right) is equal to

Options

A $\dfrac{1}{1 + {\left[ϕ\left(x\right)\right]}^{5}}$
B $\dfrac{1}{1 + {\left[f\left(x\right)\right]}^{5}}$
C $1 + {\left[ϕ\left(x\right)\right]}^{5}$
D $1 + f ( x )$

Correct answer

C. $1 + {\left[ϕ\left(x\right)\right]}^{5}$

Step-by-step solution

We have,

ϕ\left(x\right) = f^{- 1} ( x )

⇒ x = f [ ϕ\left(x\right) ]

On differentiating both sides w.r.t.

x

, we get

1 = f^{'}\left[ϕ\left(x\right)\right] . {ϕ}^{'} ( x )

⇒ {ϕ}^{'}\left(x\right) = \dfrac{1}{f^{'}\left[ϕ\left(x\right)\right]}

...(i)

Since,

f^{′}\left(x\right) = \dfrac{1}{1 + x^{5}}

(given)

f^{'}\left(ϕ\left(x\right)\right) = \dfrac{1}{1 + \left[ϕ\left(x\right)\right]5}

From Equation (i),

{ϕ}^{'}\left(x\right) = \dfrac{1}{f^{'} [ ϕ\left(x\right) ]} = 1 + \left[ϕ\left(x\right)\right]5

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