Question

y=(log x)^x then dy /dx is?​

Answer 1

$</p><p></p><p> \sf \rightarrow \quad y = {log \: x}^{x} \\ \\ \sf \rightarrow \quad y = x \: log \: x \\ \\ \sf \rightarrow \quad \frac{dy}{dx} = \frac{d}{dx} \bigg(x \cdot log \: x \bigg) \\ \\ \sf \rightarrow \quad \frac{dy}{dx} = \frac{d}{dx} \bigg( x \cdot \dfrac{ ln \: x }{ln \: 10} \bigg) \\ \\ \sf \rightarrow \quad \frac{dy}{dx} = \dfrac{1}{ln \: 10} \bigg( x \cdot ln \: x\bigg) \\ \\ \sf \rightarrow \quad \frac{dy}{dx} = \frac{1}{ln \: 10} \bigg \{ \frac{dx}{dx}ln \: x + \frac{d(ln \: x)}{dx} x \bigg \} \\ \\ \sf \rightarrow \quad \frac{dy}{dx} = \frac{1}{ln \: 10} \bigg( ln \: x + \frac{1}{ \cancel{x}} \cdot \cancel{x} \bigg) \\ \\ \sf \rightarrow \quad \frac{dy}{dx} = \dfrac{1 + ln \: x}{ln \: 10}</p><p></p><p>$