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  • Let  be the inverse of an invertible function <img alt="f(x)" src="https://learn.careers360.com/latex-image/?f%2

Let g(x) be the inverse of an invertible function f(x) which is differentiable for all real x, then {g}''(f(x)) equals to

Option: 1

\frac{-{f}''(x)}{({f}'(x))^{3}}
 

 


Option: 2

\frac{{f}'(x){f}''(x)-({f}'(x))^{3}}{{f}'(x)}

 


Option: 3

\frac{{f}'(x){f}''(x)-({f}'(x))^{2}}{({f}'(x))^{2}}


Option: 4

None of these


Answers (1)

best_answer

 

Quotient Rule of differentiation -

\frac{d}{dx}\:\:\frac{f(x)}{g(x)}=\frac{g(x).f'(x)-f(x).g'(x)}{[g(x)]^{2}}


\frac{d}{dx}\:\:\frac{u}{v}=\frac{v.\frac{du}{dx}-u.\frac{dv}{dx}}{v^{2}}

-

 

 

Given that   g^{-1}(x)=f(x)

\Rightarrow x=g\left ( f(x) \right )\:\:\:or\:\:\:g'\left ( f(x) \right )f'(x)=1

\Rightarrow g ' \left ( f(x) \right )=\frac{1}{f'(x)}

\Rightarrow g'' \left ( f(x) \right ) . f ' (x)=\frac{-f''(x)}{\left ( f'(x) \right )^{2}}

\Rightarrow g'' \left ( f(x) \right ) = - \frac{f''(x)}{\left ( f'(x) \right )^{3}}

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Rishi

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