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JEE MainMathematicsModerateDifferentiation

If f(x)=log⁡x(lnx),f(x) = \log_{x}(lnx), then f' (x) at x=e is equal to

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f(x)f(x)= log_x(lnx)
⇒\Rightarrow f(x)=ln(lnx)lnxf(x) = \frac{ln(lnx)}{lnx} ⇒\Rightarrow
f′(x)=lnx(1lnx⋅1x)−1xln(lnx)(lnx)2f^{'}(x) = \frac{lnx\left( \frac{1}{lnx} \cdot \frac{1}{x} \right) - \frac{1}{x}ln(lnx)}{(lnx)^{2}}
⇒\Rightarrow f′(e)=1e−01=1/ef^{'}(e) = \frac{\frac{1}{e} - 0}{1} = 1/e

Source: Grass-G / M-12 Mathematics Paper 2 Nov.docx

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