# 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.

Source: https://practicecalculus.com/rules/constant-rule
Updated: 2026-11-03

$$ \frac{d}{dx}\, c = 0 $$

## 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

## Key takeaways

- The derivative of any constant is $0$: $\frac{d}{dx}\,c = 0$.
- Numbers like $\pi$, $e^3$ and $\ln 2$ are constants, so their derivatives are $0$ too.
- A constant added to a function disappears when you differentiate. A constant multiplied by a function stays.

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 $c$:

$$\frac{d}{dx}\, c = 0$$

"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 $\ln 5$.

```graph
title: 'Graph of the constant function y = 3 and its derivative y = 0'
caption: 'The line $y = 3$ is horizontal, so its slope at every point is $0$. The derivative is the line $y = 0$.'
x: [-4, 4]
y: [-1, 4]
functions:
  - expr: '3'
    label: y = 3
  - expr: '0'
    label: y = 0 (derivative)
    dashed: true
```

## 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](/rules/sum-and-difference-rule), each term is differentiated separately and the constant term drops out:

$$\frac{d}{dx}\left(x^2 + 4x + 9\right) = 2x + 4 + 0 = 2x + 4$$

**Example: a parameter.** If $a$ and $b$ are constants, then

$$\frac{d}{dx}\left(a^2 + bx\right) = 0 + b = b$$

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](/blog/limit-definition-of-the-derivative) with $f(x) = c$:

$$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = \lim_{h \to 0} \frac{c - c}{h} = \lim_{h \to 0} 0 = 0$$

You can also see it as a special case of the [power rule](/rules/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](/rules/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:

$$\int 2x\,dx = x^2 + C$$

See [antiderivatives](/blog/antiderivatives) for more on why $+\,C$ is required.

## Common mistakes

1. **Treating constant expressions as variables.** $\frac{d}{dx}\,e^2 = 0$, not $2e$ or $e^2$. The exponent is a number, not $x$.
2. **Zeroing out a constant multiple.** $\frac{d}{dx}\,4x = 4$, not $0$. Only a constant on its own, or added, becomes $0$.
3. **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](/rules/power-rule) and the [sum rule](/rules/sum-and-difference-rule). See the full guide to [the power rule](/blog/power-rule) for practice with all three together.

## Frequently asked questions

### What is the derivative of a constant?

Zero. A constant function has a horizontal graph, and a horizontal line has slope $0$ at every point.

### What is the derivative of pi?

$\pi$ is a number, about $3.14159$, so $\frac{d}{dx}\,\pi = 0$. The same goes for $\pi^2$, $e$, $e^5$ and $\sqrt{2}$.

### Why is the derivative of 5x not 0?

$5x$ is a constant times $x$, not a constant. By the constant multiple rule its derivative is $5 \cdot 1 = 5$.

### Why do antiderivatives have a + C?

Because the derivative of every constant is $0$, functions such as $x^2$, $x^2 + 3$ and $x^2 - 10$ all have derivative $2x$. The $+\,C$ stands for that whole family.