Derivative of mod x | Mod x Derivative

Derivative of absolute value of x. The derivative of mod x is denoted by d/dx(|x|) and it is equal to x/|x| for all nonzero values of x. In this post, we will learn how to differentiate modulus x.

Recall that mod x is defined as below.

$|x|=\{\begin{array}{ll}x,& \text{ if }x\ge 0\\ -x,& \text{ if }x<0\end{array}$

Derivative of |x|

The derivative of |x|, that is, the derivative of modulus x, denoted by d/dx(|x|), is given as follows:

${\displaystyle \frac{d}{dx}}(|x|)={\displaystyle \frac{x}{|x|}}$ for $x\ne 0$.

How to Differentiate mod x?

Let us now find the derivative of mod x.

Question: Find ${\displaystyle \frac{d}{dx}}(|x|)$.

Answer:

Case 1: First we assume that x>0.

Thus, by the above definition of |x|, we have |x|=x.

∴ d/dx(|x|) = d/dx(x) = 1

Case 2: First we assume that x<0.

So, again by the above definition of |x|, we have |x|=-x.

In this case, d/dx(|x|) = d/dx(-x) = -1

Thus, we obtain that

${\displaystyle \frac{d}{dx}}(|x|)=\{\begin{array}{ll}1,& \text{ if }x>0\\ -1,& \text{ if }x<0\end{array}$

Combining both cases, we get that

${\displaystyle \frac{d}{dx}}(|x|)={\displaystyle \frac{x}{|x|}}$ for $x\ne 0$.

NOTE: The function f(x)=|x| is not differentiable. For a proof, CLICK HERE.

More Derivatives: Derivative of mod sinx

Derivative of mod cosx

Remark:

1. The derivative of mod x at x=a>0 is 1.
2. The derivative of mod x at x=a<0 is -1.
3. The derivative of mod x at x=0 does not exist.

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