Find the derivative: frac{d}{dx}(x^{log_e x}) | vedclass

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Question

(A) Let $y = x^{\log_e x}$.Taking the natural logarithm on both sides:$\log_e y = \log_e(x^{\log_e x})$$\log_e y = (\log_e x)(\log_e x) = (\log_e x)^2

Vedclass · Accepted Answer

$2x^{(\log_e x - 1)} \cdot \log_e x$

Vedclass · Answer

(A) Let $y = x^{\log_e x}$.Taking the natural logarithm on both sides:$\log_e y = \log_e(x^{\log_e x})$$\log_e y = (\log_e x)(\log_e x) = (\log_e x)^2$.Differentiating both sides with respect to $x$:$\frac{1}{y} \frac{dy}{dx} = 2(\log_e x) \cdot \frac{d}{dx}(\log_e x)$$\frac{1}{y} \frac{dy}{dx} = 2(\log_e x) \cdot \frac{1}{x}$$\frac{dy}{dx} = y \cdot \frac{2 \log_e x}{x}$Substituting $y = x^{\log_e x}$:$\frac{dy}{dx} = x^{\log_e x} \cdot \frac{2 \log_e x}{x}$$\frac{dy}{dx} = 2 x^{(\log_e x - 1)} \log_e x$.

Find the derivative: $\frac{d}{dx}(x^{\log_e x})$

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