Constant Rule (Derivative)
The derivative of any constant is zero.
Formula
Variables
Example
d/dx (7) = 0 — a constant has zero slope everywhere.
How it works
A constant does not change as x changes, so its rate of change is always 0. In derivative terms, the graph of y = c is a horizontal line, and horizontal lines have slope 0 everywhere. This comes directly from the limit definition of the derivative: the difference in function values is always 0, so the quotient stays 0.
When to use it
Use this rule whenever you are differentiating a number by itself, such as 12, -3, or π. It also helps inside larger expressions, since any standalone constant term disappears when you take the derivative.
Worked example, step by step
Find the derivative of f(x) = 15. Using the constant rule: d/dx(15) = 0 So, f'(x) = 0 Final answer: 0
Common mistakes to avoid
- thinking the derivative of a constant is the constant itself
- forgetting that a constant term vanishes when differentiating a sum like x^2 + 5
- treating a constant as if it depends on x
Did You Know?
The derivative of any constant is zero — a flat line has no slope.
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