Let $\phi(x)$ be the inverse of the func

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Target Exam

CUET

Subject

-- Mathematics - Section B1

Chapter

Relations and Functions

Question:

Let $\phi(x)$ be the inverse of the function f(x) and $f'(x)=\frac{1}{1+x^5}$, then $\frac{d}{dx}\phi(x)$ is

Options:

$\frac{1}{1+(\phi(x))^5}$

$\frac{1}{1+(f(x))^5}$

$1+(\phi(x))^5$

$1+(f(x))^5$

Correct Answer:

$1+(\phi(x))^5$

Explanation:

$\phi(x)=f^{-1}(x)$

$⇒x=f(\phi(x))⇒1=(f'(\phi(x))).\phi'(x)$

$⇒\phi'(x)=\frac{1}{f'(\phi(x))}⇒f'(x)=\frac{1}{1+x^5}⇒f'(\phi(x))=\frac{1}{1+(\phi(x))^5}$

or $\phi'(x)=\frac{1}{f'(\phi(x))}=1+(\phi(x))^5$