TS EAMCET · Maths · Differentiation
If \(y=x^{\log x}+(\log x)^x, x>1\) then \(\left(\frac{d y}{d x}\right)_{x=e}=\)
- A \(0\)
- B \(1\)
- C \(2\)
- D \(3\)
Answer & Solution
Correct Answer
(D) \(3\)
Step-by-step Solution
Detailed explanation
\(y=u+v\), where \(u=x^{\log x}\) and \(v=(\log x)^x\). \(\log u = (\log x)^2 \implies \frac{du}{dx} = x^{\log x} \cdot \frac{2\log x}{x}\) \(\left(\frac{du}{dx}\right)_{x=e} = e^{\log e} \cdot \frac{2\log e}{e} = e^1 \cdot \frac{2 \cdot 1}{e} = 2\)…
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