What is the derivative of an absolute value?

The derivative of an absolute value function can be understood by considering its definition and behavior.

Let's break it down:

The absolute value function is defined as:

|x| =

x when x ≥ 0,
-x when x < 0.
We want to find the derivative of the function f(x) = |x|. The derivative of an absolute value function depends on whether x is positive, negative, or zero. Here's how we handle each case:

For x > 0:
When x is positive, |x| = x.
So, the derivative of f(x) = x is simply:
f'(x) = 1.

For x < 0:
When x is negative, |x| = -x.
So, the derivative of f(x) = -x is:
f'(x) = -1.

At x = 0:
The derivative at x = 0 does not exist. This is because the absolute value function has a sharp corner at x = 0, and the left-hand and right-hand derivatives are not equal.

From the left, as x approaches 0 from negative values, the derivative approaches -1.
From the right, as x approaches 0 from positive values, the derivative approaches 1.
Since these two derivatives do not match at x = 0, the derivative at this point is undefined.
Thus, the derivative of f(x) = |x| is:

f'(x) = 1, for x > 0,
f'(x) = -1, for x < 0,
f'(x) does not exist, for x = 0.