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JEE Main & Advanced Mathematics Differentiation Question Bank Differentiation by Substitution

  • question_answer
    If \[f(x)={{\tan }^{-1}}\left\{ \frac{\log \left( \frac{e}{{{x}^{2}}} \right)}{\log (e{{x}^{2}})} \right\}+{{\tan }^{-1}}\left( \frac{3+2\log x}{1-6\log x} \right)\], then \[\frac{{{d}^{n}}y}{d{{x}^{n}}}\] is \[(n\ge 1)\]

    A)            \[{{\tan }^{-1}}\{{{(\log x)}^{n}}\}\]

    B)            0

    C)            \[\frac{1}{2}\]

    D)            None of these

    Correct Answer: B

    Solution :

               We have \[y={{\tan }^{-1}}\left( \frac{\log e-\log {{x}^{2}}}{\log e+\log {{x}^{2}}} \right)+{{\tan }^{-1}}\left( \frac{3+2\log x}{1-6\log x} \right)\]            \[={{\tan }^{-1}}\left( \frac{1-2\log x}{1+2\log x} \right)+{{\tan }^{-1}}\left( \frac{3+2\log x}{1-6\log x} \right)\]            \[={{\tan }^{-1}}1-{{\tan }^{-1}}(2\log x)+{{\tan }^{-1}}3+{{\tan }^{-1}}(2\log x)\]            \[\Rightarrow y={{\tan }^{-1}}1+{{\tan }^{-1}}3\Rightarrow \frac{dy}{dx}=0\Rightarrow \frac{{{d}^{n}}y}{d{{x}^{n}}}=0.\]


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