If \(y = x^{\log_e x}\), \(x \gt 0\), then \(\displaystyle\frac{dy}{dx}\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{2x^{\log_e x}\log_e x}{x}\)
The power contains x, so use logarithmic differentiation:
\(\log_e y = \log_e x \cdot \log_e x = (\log_e x)^{2}\)
Differentiating with respect to x by the chain rule:
\( \displaystyle \frac{1}{y}\frac{dy}{dx} = 2\log_e x \cdot \frac{1}{x} \)
\( \displaystyle \frac{dy}{dx} = \frac{2x^{\log_e x}\log_e x}{x} \)