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JEE Main & Advanced Mathematics Differentiation Question Bank Derivative at a point Standard Differentiation

  • question_answer
    \[\frac{d}{dx}\left[ \log \left\{ {{e}^{x}}{{\left( \frac{x-2}{x+2} \right)}^{3/4}} \right\} \right]\] equals to          [RPET 2001]

    A)            1

    B)            \[\frac{{{x}^{2}}+1}{{{x}^{2}}-4}\]

    C)            \[\frac{{{x}^{2}}-1}{{{x}^{2}}-4}\]

    D)            \[{{e}^{x}}\frac{{{x}^{2}}-1}{{{x}^{2}}-4}\]

    Correct Answer: C

    Solution :

               Let\[y=\left[ \log \left\{ {{e}^{x}}{{\left( \frac{x-2}{x+2} \right)}^{3/4}} \right\} \right]=\log {{e}^{x}}+\log {{\left( \frac{x-2}{x+2} \right)}^{3/4}}\]\[\]            Þ  \[y=x+\frac{3}{4}\,[\log (x-2)-\log (x+2)]\]            Þ  \[\frac{dy}{dx}=1+\frac{3}{4}\,\left[ \frac{1}{x-2}-\frac{1}{x+2} \right]=1+\frac{3}{({{x}^{2}}-4)}\]            Þ  \[\frac{dy}{dx}=\frac{{{x}^{2}}-1}{{{x}^{2}}-4}\].


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