AP EAMCET · Maths · Differentiation
If \(y=(\log x)^{1 / x}+x^{\log x}\), at \(x=e, \frac{d y}{d x}=\)
- A \(2+\frac{1}{\mathrm{e}}\)
- B \(\mathrm{e}^2+\frac{1}{2}\)
- C \(\frac{1}{\mathrm{e}^2}+2\)
- D \(e+\frac{1}{e}\)
Answer & Solution
Correct Answer
(C) \(\frac{1}{\mathrm{e}^2}+2\)
Step-by-step Solution
Detailed explanation
Let \( u = (\log x)^{1/x} \) and \( v = x^{\log x} \). For \( u \): \( \ln u = \frac{1}{x} \ln(\log x) \) \( \frac{1}{u}\frac{du}{dx} = -\frac{1}{x^2}\ln(\log x) + \frac{1}{x} \frac{1}{\log x} \frac{1}{x} = \frac{1}{x^2}\left(\frac{1}{\log x} - \ln(\log x)\right) \)…
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