Question

: Find the derivatives of the function

$f\left(x\right)=x^x$

Short answer

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

Step-by-step solution

Step 1:Given information

The given information is$f\left(x\right)=x^x$

Step 2:Simplification

Given

$f\left(x\right)=y=x^x$

Taking log both sides

$\log y=\log x^x$

$\Rightarrow \log y=x\log x$ (log property $\log a^b=b\log a$)

Differentiating w.r.t x

$\Rightarrow \frac{1}{y}\frac{dy}{dx}=\frac{d}{dx}\left(x\log x\right)$

$\Rightarrow \frac{1}{y}\frac{dy}{dx}=\log x+x.\frac{1}{x}$ (product rule of differentiation)

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

$\Rightarrow \frac{dy}{dx}=x^x\left(\log x+1\right)$