Question

If $y=x^x$, find $\frac{dy}{dx}$.

Answer 1

Given that $y=x^x$

Now apply logarithm on both sides , we get $\ln \left(y\right)=x\ln \left(x\right)$

Differentiate both sides w.r.t $x$, we get

$\frac{1}{y}\times \frac{dy}{dx}=\ln \left(x\right)+x\times \frac{1}{x}=1+\ln \left(x\right)$

$⟹\frac{dy}{dx}=y(1+\ln (x\left)\right)=x^x(1+\ln (x\left)\right)$

Therefore the value of $\frac{dy}{dx}$ is $x^x(1+\ln (x\left)\right)$