Derivative of Absolute Value Function

Absolute value is defined as the value which is equals to both positive value and negative value. Absolute value makes the negative number into a positive number. In absolute value both the real number and its negative have the same value. Absolute value is represented by '||'. This symbol is known as bar. Sometimes the absolute value is also written as ‘abs( )'.

A real value absolute function is a continuous function for every Point. We can say that Absolute Value Function is differentiable at each and every Point except point at x = 0. This absolute function is not defined at point x = 0. Absolute value Functions have this property that they are idempotent in nature, which means that if we apply value many times we get same value that we get first time.

The formula to calculate the derivative of the absolute value of a function is given as,
$\frac{d}{dx}$f(x) = $\frac{f(x)}{|f(x)|}$ . f'(x)

Some properties of absolute value are given below:

1. Non-Negativity Property: |a| $\geq$ 0
2. Symmetric Property: |-a| = a
3. Additive Property: |a + b| $\leq$ |a| + |b|
4. Subtractive Property: |a - b| $\geq$ |a| - |b|
5. Multiplicative Property: |ab| = |a| |b|
6. Division Property: $|\frac{a}{b}|$ = $\frac{|a|}{|b|}$
7. Idempotence Property: ||a|| = |a|