Derivative of floor(x)

Derivative

floor(x) → undefined at integers

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

floor(x) is constant between integers and jumps by 1 at each integer. On each open interval (n, n+1) the slope is 0. At each integer the jump means the derivative does not exist: the difference quotient does not approach a single number. This calculator will not print a derivative for floor(x), because an unevaluated diff() is not an answer. The honest result is: 0 between integers, undefined at integers.

Apply the definition as a limit to floor(x). The form to remember, already simplified, is undefined.

Between consecutive integers the derivative is 0. At each integer the jump means the derivative does not exist.

At x = 1.5, substitute into the simplified form undefined. That number is the slope of floor(x) there; a secant over a small step in x lands next to it.

Worked derivative

1. Write f(x) = floor(x).
2. Apply the definition as a limit to floor(x). The form to remember, already simplified, is undefined.
3. At x = 1.5, substitute into the simplified form undefined. That number is the slope of floor(x) there; a secant over a small step in x lands next to it.

Common mistakes

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 sign(x) — Same family of differentiation, different function.

Frequently asked questions