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JEE Main & Advanced Mathematics Applications of Derivatives Question Bank Critical Thinking

  • question_answer
     If \[y={{(x\log x)}^{\log \,\log x}}\], then \[\frac{dy}{dx}=\]                        [Roorkee 1981]

    A) \[{{(x\log x)}^{\log \log x}}\left\{ \frac{1}{x\log x}(\log x+\log \log x)+(\log \,\,\log x)\text{ }\left( \frac{1}{x}+\frac{1}{x\log x} \right)\text{ } \right\}\]         

    B) \[{{(x\log x)}^{x\log x}}\log \log x\left[ \frac{2}{\log x}+\frac{1}{x} \right]\]

    C) \[{{(x\log x)}^{x\log x}}\log \log x\left[ \frac{2}{\log x}+\frac{1}{x} \right]\]

    D) None of these

    Correct Answer: A

    Solution :

    • \[y={{(x\log x)}^{\log \log x}}\]                   
    • Þ  \[\log y=\log \log x[\log x+\log \log x]\]                   
    • Þ \[\frac{1}{y}\frac{dy}{dx}=\frac{1}{x\log x}(\log x+\log \log x)+\log \log x\left( \frac{1}{x}+\frac{1}{x\log x} \right)\]                   
    • Þ  \[{{s}^{2}}=a{{t}^{2}}+bt+c\].


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