Constant Rule for Derivatives: Why the Derivative of c Is 0
The constant rule says the derivative of any constant is 0. The formula, a one-line proof, examples with π and e, and the mistakes to avoid.
Derivative rules Updated 3 min read
At a glance
- Applies to
- Any expression that does not depend on x, such as 7, \pi, e^2 or \ln 5
- In words
- A constant never changes, so its rate of change is zero
- Graph meaning
- The graph of y = c is a horizontal line, which has slope 0 everywhere
- Integral version
- \int 0\,dx = C, which is why antiderivatives end with +\,C
- Related rules
- Power rule, constant multiple rule, sum rule
On this page
The constant rule says that the derivative of a constant is zero. A constant never changes, so its rate of change is 0 everywhere. It is the simplest derivative rule, and it quietly appears in almost every derivative you compute.
The constant rule formula
For any constant c:
"Constant" means anything that does not depend on the variable you are differentiating with respect to. That includes ordinary numbers, but also expressions like \pi^2, e^3, \sqrt{7}, \sin 1 and \ln 5.
Examples
| Function | Why it is constant | Derivative |
|---|---|---|
| 7 | A number | 0 |
| -12.5 | A number | 0 |
| \pi | A number, about 3.14 | 0 |
| e^3 | No x anywhere | 0 |
| \ln 5 | No x anywhere | 0 |
| \sin\frac{\pi}{6} | Equals \frac{1}{2} | 0 |
Example: a constant term in a polynomial. With the sum rule, each term is differentiated separately and the constant term drops out:
Example: a parameter. If a and b are constants, then
The a^2 looks like a power, but it does not depend on x, so it is a constant.
Why the constant rule works
Use the limit definition of the derivative with f(x) = c:
You can also see it as a special case of the power rule: c = c\,x^0, and \frac{d}{dx}\,c\,x^0 = c \cdot 0 \cdot x^{-1} = 0 (for x \ne 0).
Constants added vs constants multiplied
This is where the constant rule meets the constant multiple rule:
| Expression | Derivative | Reason |
|---|---|---|
| x^3 + 5 | 3x^2 | An added constant vanishes |
| 5x^3 | 15x^2 | A multiplied constant stays |
| 5 | 0 | Constant rule |
| 5x | 5 | Constant multiple of x |
Shifting a graph up or down by adding a constant does not change its slope anywhere, which is exactly why added constants disappear. Stretching a graph by a factor multiplies every slope by that factor, which is why multiplied constants stay.
The constant rule in reverse
Because every constant has derivative 0, many functions share the same derivative: x^2, x^2 + 1 and x^2 - 8 all differentiate to 2x. So when you integrate, you must add an arbitrary constant:
See antiderivatives for more on why +\,C is required.
Common mistakes
- Treating constant expressions as variables. \frac{d}{dx}\,e^2 = 0, not 2e or e^2. The exponent is a number, not x.
- Zeroing out a constant multiple. \frac{d}{dx}\,4x = 4, not 0. Only a constant on its own, or added, becomes 0.
- Confusing which variable is changing. \frac{d}{dx}\,t^2 = 0 when t is a constant, but \frac{d}{dt}\,t^2 = 2t. Always check the variable in the denominator of \frac{d}{d\square}.
The constant rule is usually combined with the power rule and the sum rule. See the full guide to the power rule for practice with all three together.
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