Question

Derivative of mod $x$ is

Answer 1

Step-1: Simplify the given data.

Let $y=\left|x\right|$

$\Rightarrow y=\left\{\begin{array}{ll}x,& \mathrm{if}x>0\\ -x,& \mathrm{if}x<0\end{array}\right.$

mod of $x$can also write as $\left|x\right|=\sqrt{x^2}$

$\Rightarrow y={\left(x^2\right)}^{\frac{1}{2}}$

Step-2: Differentiate with respect to $x$.

$$\begin{gathered} \Rightarrow \frac{dy}{dx}=\frac{1}{2}{\left(x^2\right)}^{\left(-\frac{1}{2}\right)}\times 2x \\ \Rightarrow \frac{dy}{dx}=\frac{x}{\sqrt{x^2}} \\ \Rightarrow \frac{dy}{dx}=\frac{x}{\left|x\right|}x\ne 0 \\ \Rightarrow \frac{dy}{dx}=\left\{\begin{array}{ll}-1,& x<0\\ 1,& x>0\end{array}\right.x\ne 0 \end{gathered}$$

Therefore, derivative of mod $x$ is $-1$ when $x<0$ and $1$ when $x>0$ and not differentiable at $x=0$.