Question

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

Answer 1

$$\begin{gathered} \text{Case 1:} \\ x>0 \\ \left|x\right|=x \\ \left(x+\left|x\right|\right)\left|x\right| \\ =\left(x+x\right)x \\ =2x^2 \\ \frac{d}{dx}\left[\left(x+\left|x\right|\right)\left|x\right|\right]=\frac{d}{dx}\left(2x^2\right)=4x\left(1\right) \\ \text{Case 2:} \\ x<0 \\ \left|x\right|=-x \\ \left(x+\left|x\right|\right)\left|x\right| \\ =\left(x-x\right)x \\ =0 \\ \frac{d}{dx}\left[\left(x+\left|x\right|\right)\left|x\right|\right]=\frac{d}{dx}\left(0\right)=0\left(2\right) \\ \text{From (1) and (2), we have:} \\ \frac{d}{dx}\left[\left(x+\left|x\right|\right)\left|x\right|\right]=\left\{\begin{array}{l}4x,\mathrm{if}x>0\\ 0,\mathrm{if}x<0\end{array}\right. \end{gathered}$$