Question

what is the derivative of x^x?

Answer 1

Let \(y = x^x\)
\(\implies ln(y) = x*ln(x)\)
\(\implies \frac{d}{dx} ln (y) = \frac{d}{dx}[x * lnx]\)
\(\implies \frac{d}{dy} ln(y) * \frac{dy}{dx} = x*\frac{d}{dx}ln(x) + ln(x) * \frac{d}{dx}x\)
(chain rule) ( product rule )
\(\implies \frac{1}{y}* \frac{dy}{dx} = \frac{x}{x} + ln(x)\)
\(\implies \frac{dy}{dx} = y(1+ln(x))\)
\(=x^x(1+ln(x))\)