Derivative of Abs(x)
Answer
Derivative of Abs(x)
- Rule Used
- Absolute Value Rule
- Steps
- Recognize the absolute value function |u| = abs(u).
- Apply the rule: |u| = u' · u/|u| (equivalently u'·sign(u) for u ≠ 0). The derivative does not exist where u = 0, because the graph has a corner.
- Here u = x, so the derivative is shown below.
- Final answer: abs(x)^(-1)*x
d/dx |x| = abs(x)^(-1)*x. The derivative exists for every x except 0, where the graph has a corner and the one-sided slopes disagree.
Instant step-by-step solution for Abs(x)
Derivative of Abs(x) = abs(x)^(-1)*x Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-abs-x
Derivative
|x| → abs(x)^(-1)*xengine: abs(x)^(-1)*x
Graph of the function and its derivative
The solid curve is f. The dashed curve is f′. Vertical gaps are asymptotes or jumps, not extra zeros.
Why the derivative has this form
Absolute value is the line y = x for x > 0 and y = −x for x < 0. The slope is therefore +1 on the right and −1 on the left, which is x/|x|, the same as the sign of x. The engine writes that as x/abs(x). At x = 0 the left-hand slope and the right-hand slope disagree, so the derivative does not exist at the corner, even though the function is continuous there.
Apply the definition as a limit to |x|. The form to remember, already simplified, is abs(x)^(-1)*x.
The derivative exists for every x except 0, where the graph has a corner and the one-sided slopes disagree.
At x = 2, substitute into the simplified form abs(x)^(-1)*x. That number is the slope of |x| there; a secant over a small step in x lands next to it.
Worked derivative
- Write f(x) = |x|.
- Apply the definition as a limit to |x|. The form to remember, already simplified, is abs(x)^(-1)*x.
- At x = 2, substitute into the simplified form abs(x)^(-1)*x. That number is the slope of |x| there; a secant over a small step in x lands next to it.
Common mistakes
Answering 1
Answering 1 describes only the right-hand ray. For negative x the graph is falling, and the slope is −1.
Claiming |x| has a derivative at the bad point
Corners and jumps are exactly where a formula that works on each piece fails. State the pieces and then state the exception.
Related rule and examples
- Definition of the derivative — The rule page lists this example and the others in the same family.
- Derivative of sign(x) — Same family of differentiation, different function.
- Derivative of floor(x) — Same family of differentiation, different function.
Frequently asked questions
What is the derivative of |x|?
abs(x)^(-1)*x. Absolute value is the line y = x for x > 0 and y = −x for x < 0. The slope is therefore +1 on the right and −1 on the left, which is x/|x|, the same as the sign of x. The engine writes that as x/abs(x). At x = 0 the left-hand slope and the right-hand slope disagree, so the derivative does not exist at the corner, even though the function is continuous there.
Which wrong answer shows up for |x|?
Answering 1 describes only the right-hand ray. For negative x the graph is falling, and the slope is −1.
Where is the derivative of |x| defined?
The derivative exists for every x except 0, where the graph has a corner and the one-sided slopes disagree.