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JEE Main & Advanced Mathematics Differentiation Question Bank Differentiation of implicit function Parametric

  • question_answer
    Let g (x) be the inverse of an invertible function \[f(x)\] which is differentiable at x = c, then \[g'(f(c))\]equals

    A)            \[f'(c)\]

    B)            \[\frac{1}{f'(c)}\]

    C)            \[f(c)\]                                        

    D)            None of these

    Correct Answer: B

    Solution :

               Since \[g(x)\] is the inverse of function \[f(x)\], therefore \[gof(x)=I(x)\] for all x.            Now \[gof(x)=I(x),\ \ \forall x\] \[\]    \[\Rightarrow gof(x)=x,\ \ \forall x\]\[\Rightarrow \]\[(gof)'(x)=1,\ \ \forall x\]                    Þ  \[g'(f(x))f'(x)=1,\ \ \forall x\]                              (using chain rule)            Þ \[g'(f(x))=\frac{1}{f'(x)},\ \ \forall x\Rightarrow g'(f(c))=\frac{1}{f'(c)}\](putting x=c)


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