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

  • question_answer
    If \[y={{x}^{{{x}^{x......\infty }}}}\], then \[\frac{dy}{dx}=\] [UPSEAT 2004; DCE 2000]

    A) \[\frac{{{y}^{2}}}{x(1+y\log x)}\]

    B) \[\frac{{{y}^{2}}}{x(1-y\log x)}\]

    C) \[\frac{y}{x(1+y\log x)}\]

    D) \[\frac{y}{x(1-y\log x)}\]

    Correct Answer: B

    Solution :

    • \[y={{x}^{{{x}^{x.......\infty }}}}\]Þ\[y={{x}^{y}}\]Þ\[\log y=y\log x\]            
    • Therefore, on differentiating \[\frac{dy}{dx}=\frac{{{y}^{2}}}{x(1-y\log x)}\].


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