The differential of the absolute value function, f(x) = |x|, is given by x/|x| for x ≠ 0. It is undefined at x = 0.
Understanding the Absolute Value Function
The absolute value function, denoted as |x|, returns the non-negative value of a real number x. It can be defined piecewise as:
|x| =
- x, if x ≥ 0
- -x, if x < 0
Differentiation
To find the derivative (and hence the differential) of |x|, we need to consider the two cases separately:
-
Case 1: x > 0
In this case, |x| = x. The derivative of x with respect to x is 1. -
Case 2: x < 0
In this case, |x| = -x. The derivative of -x with respect to x is -1. -
Case 3: x = 0
The derivative is undefined at x = 0, because the left-hand limit and right-hand limit of the derivative don't match.
We can express this derivative concisely as:
f'(x) =
- 1, if x > 0
- -1, if x < 0
- Undefined, if x = 0
Alternative Representation
The derivative of the absolute value function can also be written as:
f'(x) = x/|x|, for x ≠ 0
This is because:
- If x > 0, then x/|x| = x/x = 1
- If x < 0, then x/|x| = x/(-x) = -1
This representation clearly shows that the derivative is not defined at x = 0 because division by zero would occur.
Differential
The differential of a function f(x) is given by df = f'(x) dx, where f'(x) is the derivative of f(x) and dx is the differential of x.
Therefore, the differential of the absolute value function is:
df = (x/|x|) dx, for x ≠ 0
Summary
The differential of the absolute value function |x| is (x/|x|) dx for x ≠ 0 and is undefined at x = 0. The function is not differentiable at x=0 because of the sharp turn (a cusp) in the graph of the function at that point.
