Derivative of sign(x)
Answer
Derivative of sign(x)
- Rule Used
- Power Rule
- Steps
- Identify the expression as a power function and apply the power rule. A rational exponent is allowed: ^(p/q) = (p/q)·.
- Final answer: 0
d/dx sign(x) = 0. The derivative is 0 for x < 0 and for x > 0. It does not exist at x = 0, where sign(x) jumps.
Instant step-by-step solution for sign(x)
Derivative of sign(x) = 0 Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-sign-x
Derivative
sign(x) → 0engine: 0
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
sign(x) is −1 on the negatives, undefined or zero at the origin depending on convention, and +1 on the positives. On each open half-line the function is constant, so the slope is 0. The engine reports 0, and that answer is right only away from zero. At x = 0 the function jumps, and a jump has no derivative. Do not read the printed 0 as a derivative at the origin.
Apply the definition as a limit to sign(x). The form to remember, already simplified, is 0.
The derivative is 0 for x < 0 and for x > 0. It does not exist at x = 0, where sign(x) jumps.
At x = 2, substitute into the simplified form 0. That number is the slope of sign(x) there; a secant over a small step in x lands next to it.
Worked derivative
- Write f(x) = sign(x).
- Apply the definition as a limit to sign(x). The form to remember, already simplified, is 0.
- At x = 2, substitute into the simplified form 0. That number is the slope of sign(x) there; a secant over a small step in x lands next to it.
Common mistakes
Answering 1
Answering 1 confuses the value of the function on the right with its slope. The pieces are horizontal, so the slope is 0, not 1.
Claiming sign(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 |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 sign(x)?
0. sign(x) is −1 on the negatives, undefined or zero at the origin depending on convention, and +1 on the positives. On each open half-line the function is constant, so the slope is 0. The engine reports 0, and that answer is right only away from zero. At x = 0 the function jumps, and a jump has no derivative. Do not read the printed 0 as a derivative at the origin.
Which wrong answer shows up for sign(x)?
Answering 1 confuses the value of the function on the right with its slope. The pieces are horizontal, so the slope is 0, not 1.
Where is the derivative of sign(x) defined?
The derivative is 0 for x < 0 and for x > 0. It does not exist at x = 0, where sign(x) jumps.