Question

Write the value of $\frac{d}{dx}\left(x\left|x\right|\right)$.

Answer 1

$$\begin{gathered} \text{Case 1:} \\ x>0 \\ \left|x\right|=x \\ \text{Thus, we have:} \\ \frac{d}{dx}\left(x\left|x\right|\right)=\frac{d}{dx}\left(x.x\right)=\frac{d}{dx}\left(x^2\right)=2x\left(1\right) \\ \text{Case 2:} \\ x<0 \\ \left|x\right|=-x \\ \text{Thus, we have:} \\ \frac{d}{dx}\left(x\left|x\right|\right)=\frac{d}{dx}\left(x.\left(-x\right)\right)=\frac{d}{dx}\left(-x^2\right)=-2x\left(2\right) \\ \text{From (1) and (2), we have:} \\ \frac{d}{dx}\left(x\left|x\right|\right)=\left\{\begin{array}{l}2x,\mathrm{if}x>0\\ -2x,\mathrm{if}x<0\end{array}\right. \end{gathered}$$