# Constant Multiple Rule for Derivatives: Formula and Examples

> The constant multiple rule: the derivative of c·f(x) is c·f'(x). The formula, worked examples, a quick proof and the mistakes students make with it.

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

$$ \frac{d}{dx}\,[c\,f(x)] = c\,f'(x) $$

## At a glance

- **Applies to:** A constant $c$ multiplied by any differentiable function $f$
- **In words:** Keep the constant, differentiate the function
- **Includes division:** Dividing by a constant $k$ is multiplying by $\frac{1}{k}$
- **Integral version:** $\int c\,f(x)\,dx = c \int f(x)\,dx$
- **Related rules:** Power rule, sum rule, product rule

## Key takeaways

- A constant factor passes straight through the derivative: $(cf)' = cf'$.
- Dividing by a constant counts too: $\frac{d}{dx}\frac{x^4}{8} = \frac{4x^3}{8} = \frac{x^3}{2}$.
- You do not need the product rule when one factor is a constant.

The **constant multiple rule** says that a constant factor can be pulled out in front of a derivative. To differentiate $c\,f(x)$, keep the $c$ and differentiate $f$. Together with the [power rule](/rules/power-rule) and the [sum rule](/rules/sum-and-difference-rule), it lets you differentiate any polynomial at sight.

## The constant multiple rule formula

If $c$ is a constant and $f$ is differentiable, then

$$\frac{d}{dx}\,[c\,f(x)] = c\,f'(x)$$

Geometrically, multiplying a function by $c$ stretches its graph vertically by a factor of $c$. Every slope gets multiplied by $c$ too.

```graph
title: 'Graphs of y = x² and y = 3x² with tangent lines at x = 1'
caption: 'At $x = 1$ the slope of $x^2$ is $2$. Tripling the function triples the slope to $6$.'
x: [-2, 2.5]
y: [-2, 8]
functions:
  - expr: x^2
    label: y = x²
  - expr: 3x^2
    label: y = 3x²
  - expr: 2x - 1
    label: slope 2 at x = 1
    dashed: true
  - expr: 6x - 3
    label: slope 6 at x = 1
    dashed: true
points:
  - [1, 1, '']
  - [1, 3, '']
```

## Examples

| Function | Derivative |
| --- | --- |
| $5x^3$ | $5 \cdot 3x^2 = 15x^2$ |
| $-4\sin x$ | $-4\cos x$ |
| $2e^x$ | $2e^x$ |
| $\frac{x^2}{7}$ | $\frac{2x}{7}$ |
| $\pi x$ | $\pi$ |
| $\frac{3}{x^2} = 3x^{-2}$ | $-6x^{-3} = -\frac{6}{x^3}$ |

**Example: a constant in the denominator.** The function $\frac{1}{3x}$ has a variable in the denominator, so rewrite it as $\frac{1}{3}x^{-1}$ first:

$$\frac{d}{dx}\,\frac{1}{3}x^{-1} = \frac{1}{3}\cdot(-1)x^{-2} = -\frac{1}{3x^2}$$

**Example: combined with the sum rule.** Each term keeps its coefficient:

$$\frac{d}{dx}\left(6x^4 - \frac{1}{2}x^2 + 3\cos x\right) = 24x^3 - x - 3\sin x$$

## How to recognize a constant multiple

A constant multiple is any factor that does not change when $x$ changes. Look for it in these places:

- **A number in front of a term**, such as the $5$ in $5x^3$ or the $-4$ in $-4\sin x$.
- **A number in the denominator**, such as the $7$ in $\frac{x^2}{7}$. Dividing by $7$ is the same as multiplying by $\frac{1}{7}$.
- **A named constant**, such as $\pi$, $e^2$ or $\ln 3$. These look like functions or powers, but they contain no $x$, so they are just numbers.
- **A letter that stands for a fixed value**, such as $a$, $k$ or $m$ in a physics formula. In $\frac{d}{dx}\,k x^2 = 2kx$, the $k$ is treated exactly like a number.

If the factor contains $x$ anywhere, it is not a constant, and you need the [product rule](/rules/product-rule) or [quotient rule](/rules/quotient-rule) instead.

## Why the constant multiple rule works

From the [limit definition of the derivative](/blog/limit-definition-of-the-derivative), the constant factors out of the difference quotient and then out of the limit:

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

The second step uses the constant multiple law for limits from the [limit laws](/rules/limit-laws).

## Linearity

The constant multiple rule and the sum rule together say that differentiation is **linear**:

$$\frac{d}{dx}\,[a\,f(x) + b\,g(x)] = a\,f'(x) + b\,g'(x)$$

for any constants $a$ and $b$. Integration has the same property, which is why $\int 5x^2\,dx = 5\int x^2\,dx$.

## Common mistakes

1. **Using the product rule when it is not needed.** For $7x^3$ the product rule gives $0 \cdot x^3 + 7 \cdot 3x^2$, which is correct but slow. Just write $21x^2$.
2. **Mixing it up with the constant rule.** $\frac{d}{dx}\,8x = 8$, not $0$. Only a constant standing alone becomes $0$.
3. **Pulling a constant out of a function's input.** $\sin(3x)$ is not $3\sin x$. A constant inside the function needs the [chain rule](/rules/chain-rule): $\frac{d}{dx}\sin(3x) = 3\cos(3x)$.
4. **Treating a variable as a constant.** In $x\,e^x$, the $x$ is not a constant, so you need the [product rule](/rules/product-rule).

For more practice combining these rules, see the full guide to [the power rule](/blog/power-rule).

## Frequently asked questions

### What is the constant multiple rule?

It says $\frac{d}{dx}[c\,f(x)] = c\,f'(x)$. A constant multiplying a function stays in front, and you differentiate only the function.

### What is the difference between the constant rule and the constant multiple rule?

The constant rule says a constant on its own has derivative $0$. The constant multiple rule says a constant multiplying a function is kept, so $\frac{d}{dx}\,5 = 0$ but $\frac{d}{dx}\,5x = 5$.

### Can you use the product rule instead?

Yes, it gives the same answer because the derivative of the constant is $0$, but it is extra work. The constant multiple rule is the shortcut.

### Does the constant multiple rule work with a constant in the denominator?

Yes. $\frac{f(x)}{k}$ is $\frac{1}{k}f(x)$, so its derivative is $\frac{f'(x)}{k}$. A variable in the denominator is different and needs the power rule or quotient rule.