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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 the function \[f(x)\] and \[f'(x)=\frac{1}{1+{{x}^{3}}}\]. Then \[{g}'(x)\] is equal to [Kurukshetra CEE 1996]

    A)            \[\frac{1}{1+{{(g(x))}^{3}}}\]

    B)            \[\frac{1}{1+{{(f(x))}^{3}}}\]

    C)            \[1+{{(g(x))}^{3}}\]

    D)            \[1+{{(f(x))}^{3}}\]

    Correct Answer: C

    Solution :

               Since \[g(x)\]is the inverse of \[f(x)\], therefore                    \[f(x)=y\]\[\Leftrightarrow g(y)=x\]                    Now, \[g'(f(x))=\frac{1}{f'(x)}\,,\,\forall x\]Þ \[g'(f(x))=1+{{x}^{3}},\ \ \forall x\]            Þ  \[g'(y)=1+{{(g(y))}^{3}}\]           [using\[f(x)=y\Leftrightarrow x=g(y)]\]                    Þ  \[g'(x)=1+{{(g(x))}^{3}}\]  (replacing y by x).


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