Question

The derivative of $f\left(x\right)=x\left|x\right|$ is

• A. $2x$
• B. $-2x$
• C. $-2x^2$
• D. $2\left|x\right|$

Answer 1

The correct option is D

$2\left|x\right|$

Explanation for the correct option:

Finding the derivative of the given function:

$$\begin{gathered} f\left(x\right)=x\left|x\right| \\ \mathrm{When}x<0,\mathrm{then}\left|x\right|=-x \\ f\left(x\right)=x\times \left(-x\right) \\ =-x^2 \\ \mathrm{When}x>0,\mathrm{then}\left|x\right|=x \\ \mathrm{Therefore}, \\ \mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}\times \mathrm{x} \\ ={\mathrm{x}}^2 \\ \mathrm{So}, \\ {\mathrm{f}}^{\text{'}}\left(\mathrm{x}\right)=\frac{\mathrm{d}}{\mathrm{dx}}\left(-{\mathrm{x}}^2\right) \\ =-2\mathrm{x},\mathrm{when}\mathrm{x}<0\to \left(1\right) \\ \mathrm{And}, \\ {\mathrm{f}}^{\text{'}}\left(\mathrm{x}\right)=\frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{x}}^2\right) \\ =2\mathrm{x},\mathrm{when}\mathrm{x}<0\to \left(2\right) \\ \mathrm{Therefore}; \\ {\mathrm{f}}^{\text{'}}\left(\mathrm{x}\right)=2\times \left|\mathrm{x}\right| \\ =2\left|\mathrm{x}\right| \end{gathered}$$

Therefore, the correct answer is option (D).