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  •  Let g(x) be the inverse of an invertible function f(x) which is differentiable at x=c, then  <img alt="\mathrm{ g^{\prime}(f(c))}" src="https://entrancecorner.oncodecogs.c

 Let g(x) be the inverse of an invertible function f(x) which is differentiable at x=c, then  \mathrm{ g^{\prime}(f(c))}  equals

Option: 1

\mathrm{f^{\prime}(c)}


Option: 2

\mathrm{\frac{1}{f^{\prime}(c)}}


Option: 3

\mathrm{f(c)}


Option: 4

none of these.


Answers (1)

best_answer

Since g(x) is the inverse of function f(x), therefore \mathrm{ g ~ of ~f(x)=I(x)}  for all x.

\mathrm{ \Rightarrow g o f(x)=x, \vee x^{\prime} \\ }
\mathrm{ \Rightarrow\left(g \circ f^{\prime}(x)=1, \vee x\right. \\ }
\mathrm{ \Rightarrow g^{\prime}(f(x)) f^{\prime}(x)=1, \vee x \\ }                 [Using chain rule]
\mathrm{ \Rightarrow g^{\prime}(f(x))=\frac{1}{f^{\prime}(x)^{\prime}} \vee x \Rightarrow g^{\prime}(f(c))=\frac{1}{f^{\prime}(c)} }   [Putting x = c ]

 

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Rakesh

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