Question

Find derivative of $x^x$.

Answer 1

Compute the required derivative:

Let us consider $y=x^x$

Taking $\log$ on both the sides, we have,

$\log \left(y\right)=x\log \left(x\right)$ $\left[\therefore \log \left(x^x\right)=x\log \left(x\right)\right]$

Differentiating the above with respect to $x$.

$$\begin{gathered} \frac{d\log \left(y\right)}{dx}=\frac{d\left(x\log \left(x\right)\right)}{dx} \\ \Rightarrow \frac{1}{y}\frac{dy}{dx}=\log \left(x\right)+\frac{1}{x}\left(x\right) \\ \Rightarrow \frac{dy}{dx}=y\left(1+\log \left(x\right)\right) \\ \Rightarrow \frac{dy}{dx}=x^x\left(1+\log \left(x\right)\right) \end{gathered}$$

Hence, the required solution is $x^x\left(1+\log \left(x\right)\right)$.