The constant function rule states that the derivative of any constant value is zero. Because a constant function always outputs the same number, it graphs as a flat, horizontal line. Therefore, its rate of change (slope) is exactly (0) at every single point.
In calculus, if f(x) = c (where c is any real number), the rule is written as:
Think of the derivative as the measure of how much a value changes as x moves. Since a constant function never changes, its instantaneous rate of change is absolutely zero.
Intuitively, this makes sense because the slope of a constant line at any point is always 0, there is no rate of change. Therefore, if you are given y = 5, y = -5 or y = 1,000,000, the derivative of all of those functions is always the same: ZERO.
References
“Constant Function – Definition, Graph, Characteristics, Examples.” CUEMATH. Accessed June 30, 2026. https://www.cuemath.com/calculus/constant-functions/.
Ming, Albert. “7 Derivative Rules You Should Know.” Medium, April 19, 2021. https://albertming88.medium.com/7-derivative-rules-you-should-know-ce405b8f9f4d.
“The Basic Rules | Calculus I.” lumen. Accessed June 30, 2026. https://courses.lumenlearning.com/calculus1/chapter/the-basic-rules/.