The Constant Multiple Rule
Textbook Reference
| Primary source | OpenStax Calculus Volume 1, Section 3.3: “Differentiation Rules” |
| Book URL | https://openstax.org/details/books/calculus-volume-1 |
Freely available and openly licensed.
Key idea
If you double the output of a function at every point, you double the rate of change at every point. Multiplying a function by a constant multiplies its derivative by the same constant. This is the constant multiple rule, and it follows directly from the constant factor rule for limits.
Prerequisite Check
Quick Reference
\[ \frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x) \]
Constants factor out of derivatives.
Key Concepts
1. Proof from the Definition
Let $g(x) = c \cdot f(x)$. Then: \[ g'(x) = \lim_{h \to 0} \frac{c \cdot f(x+h) - c \cdot f(x)}{h} = c \cdot \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = c \cdot f'(x). \]
The constant $c$ factors out of the limit because it is not affected by $h \to 0$.
2. Examples
Example 1. Differentiate $f(x) = 5x^3$.
$\dfrac{d}{dx}[5x^3] = 5 \cdot \dfrac{d}{dx}[x^3] = 5 \cdot 3x^2 = 15x^2$.
Example 2. Differentiate $g(x) = -4\sqrt{x}$.
$g'(x) = -4 \cdot \dfrac{1}{2\sqrt{x}} = \dfrac{-2}{\sqrt{x}}$.
Example 3. Differentiate $h(x) = \dfrac{x^4}{3}$.
$h'(x) = \dfrac{1}{3} \cdot 4x^3 = \dfrac{4x^3}{3}$.
3. Combined with Sum Rule and Power Rule
The constant multiple rule combines with the sum and power rules to differentiate any polynomial.
Example 4. Differentiate $p(x) = 3x^4 - 7x^2 + 2x - 9$.
$p'(x) = 3(4x^3) - 7(2x) + 2(1) - 0 = 12x^3 - 14x + 2$.
4. Geometric Meaning
Multiplying $f$ by $c$ stretches the graph vertically by a factor of $c$. This stretching makes slopes $c$ times steeper: if the original slope at $a$ was $f'(a)$, the new slope is $c \cdot f'(a)$.
Common Errors
| Error | Example | Correction |
|---|---|---|
| Applying the rule to variable factors | “$\frac{d}{dx}[x \cdot f(x)] = f(x)$” | $x$ is not a constant; use the product rule instead |
| Confusing coefficient and exponent | Differentiating $5x^3$ as $3x^3$ (moving exponent only) | $\frac{d}{dx}[5x^3] = 5 \cdot 3x^2 = 15x^2$; the 5 stays |
Leveled Practice
Level 1 -- Applying the Rule
Problem 1. Differentiate: (a) $6x^5$, (b) $-3x^2$, (c) $\frac{x^3}{4}$.
Show answer
(a) $30x^4$. (b) $-6x$. (c) $\frac{3x^2}{4}$.
Problem 2. Find $f'(x)$ for $f(x) = 8\sqrt{x}$.
Show answer
$f'(x) = 8 \cdot \frac{1}{2\sqrt{x}} = \frac{4}{\sqrt{x}}$.
Level 2 -- Polynomial Differentiation
Problem 3. Differentiate $y = 4x^3 - 10x + 7$.
Show answer
$\frac{dy}{dx} = 12x^2 - 10$.
Problem 4. Find all values of $x$ where $f'(x) = 0$ for $f(x) = 2x^3 - 18x$.
Show answer
$f'(x) = 6x^2 - 18 = 6(x^2 - 3) = 0$ gives $x = \pm\sqrt{3}$.
Level 3 -- Reasoning
Problem 5. If $f$ has $f'(2) = 5$, find the derivative of $g(x) = 7f(x)$ at $x = 2$.
Show answer
$g'(x) = 7f'(x)$, so $g'(2) = 7 \cdot 5 = 35$.
Common Misconceptions
$\frac{d}{dx}[cf(x)] = c \cdot f(x)$ (differentiation affects only the constant). The constant multiple rule says the constant passes through differentiation unchanged, and $f$ itself is differentiated: $\frac{d}{dx}[cf(x)] = c \cdot f'(x)$. A common error is keeping $f(x)$ unchanged and multiplying by $c$ again, or forgetting to differentiate $f$ at all. For $\frac{d}{dx}[5x^3]$: $c = 5$, $f(x) = x^3$, $f'(x) = 3x^2$, so the result is $5 \cdot 3x^2 = 15x^2$.
multiplying $f$ by a constant makes the derivative constant. Scaling a function vertically stretches the graph, which also scales the slope at every point by the same factor. The derivative of $5f$ is $5f'$, which is still a function that varies with $x$ (unless $f'$ happened to be constant). The constant 5 does not flatten the derivative; it amplifies it.
Mastery Checklist
Mental Model
A constant multiplier is transparent to differentiation: it stretches the graph and the slopes proportionally. If you pay twice as much for every kilometer driven, your marginal cost per kilometer doubles -- the constant multiplier scales the derivative.
Connections
Within MATH161
- Sum rule: Combining constant multiple and sum rules lets you differentiate every polynomial.
- Integration: $\int c \cdot f(x)\,dx = c \int f(x)\,dx$ is the same rule in reverse.