2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple

These derivative rules let you differentiate polynomials fast: the derivative of a constant is zero, sums and differences split apart term by term, and constant multiples pull out front. Combine them with the power rule to find the derivative of any polynomial function quickly. For AP Calculus, use these rules to show clean term-by-term work before moving into tangent lines, motion, and optimization.

Why This Matters for the AP Calculus Exam

Most derivatives you take on the AP Calculus exam start with these rules. Polynomials show up everywhere, from finding tangent line slopes to locating critical points to setting up motion problems, and you need to differentiate them without slowing down.

These rules support exam thinking in a few ways:

• On multiple-choice questions, you can differentiate a polynomial in one clean pass.
• On free-response questions, clear term-by-term work shows the structure behind your answer, which matters for clear, complete solutions.
• In later units, you will reuse these rules constantly when analyzing increasing/decreasing behavior, concavity, and instantaneous rates of change.

Getting comfortable with these rules now means you spend exam time on the harder reasoning, not the basic differentiation.

Key Takeaways

• The derivative of a constant is always 0, since a constant function never changes.
• The sum and difference rules let you differentiate each term separately, then add or subtract the results.
• The constant multiple rule lets you factor a constant out front: $\frac{d}{dx}[c \cdot g(x)] = c \cdot g'(x)$.
• These rules combine with the power rule to differentiate any polynomial term by term.
• Watch signs and coefficients carefully, since most polynomial mistakes are small arithmetic or notation slips.
• Notation like $f'(x)$ or $\frac{dy}{dx}$ should stay consistent through your work.

Key Derivative Rules

Before going further, make sure you are solid on the power rule, since you will combine it with these rules constantly. Review the power rule if you need a refresher.

The Constant Rule

The derivative of a constant is always zero. If $f(x) = c$, where $c$ is a constant, then $f'(x) = 0$.

For example, the derivative of $f(x) = 3$ is $f'(x) = 0$. A constant function is a flat horizontal line, so its slope is zero everywhere.

The Sum Rule

The derivative of a sum is the sum of the derivatives. If $f(x) = g(x) + h(x)$, then $f'(x) = g'(x) + h'(x)$.

Take $f(x) = 2x + 3$. The derivative of $2x$ is $2$, and by the constant rule the derivative of $3$ is $0$. Adding gives $2 + 0 = 2$, so $f'(x) = 2$.

The Difference Rule

The derivative of a difference is the difference of the derivatives. If $f(x) = g(x) - h(x)$, then $f'(x) = g'(x) - h'(x)$.

This works just like the sum rule, with subtraction instead of addition. For $f(x) = 2x - 3$, you get $2 - 0 = 2$.

The Constant Multiple Rule

The derivative of a constant times a function is the constant times the derivative. If $f(x) = c \cdot g(x)$, then $f'(x) = c \cdot g'(x)$.

You can see this when the derivative of $2x$ is $2 \cdot 1 = 2$. The coefficient stays put and multiplies the derivative of the rest.

Practice Problems

Work through these to lock in the rules.

Example 1

Find the derivative of $f(x) = 2x^2 + 2$.

The plus sign tells you to use the sum rule.

Step 1: Split into two pieces. Here $g(x) = 2x^2$ and $h(x) = 2$.

Step 2: Differentiate each piece and add. $g'(x) = 4x$ and $h'(x) = 0$, so $f'(x) = 4x + 0 = 4x$.

Example 2

Find the derivative of $f(x) = 100$.

Since $100$ is a constant, the constant rule gives $f'(x) = 0$.

Example 3

Find the derivative of $f(x) = 5(5x + 10)$.

The constant multiple rule fits here.

Step 1: Identify the constant and the function. Here $c = 5$ and $g(x) = 5x + 10$.

Step 2: Differentiate $g(x)$ and multiply by the constant. $g'(x) = 5$, so $f'(x) = 5 \cdot 5 = 25$.

Example 4

Find the derivative of $f(x) = 3x^3 - 6x$.

Step 1: Split using the difference rule. Here $g(x) = 3x^3$ and $h(x) = 6x$.

Step 2: Differentiate each piece and subtract. $g'(x) = 9x^2$ and $h'(x) = 6$, so $f'(x) = 9x^2 - 6$.

Combining the Power Rule with These Rules

On the AP Calculus exam, you will often need several rules at once to differentiate a polynomial. Use this approach:

1. Differentiate each term using the power rule.
2. Add or subtract the term derivatives according to the original function's signs.

Polynomial Example 1

Differentiate $f(x) = 3x^4 - 2x^3 + 5x^2 - 7x + 9$.

Step 1: Apply the power rule to each term.

$\frac{d}{dx} (3x^4) = 4 \cdot 3x^{4-1} = 12x^3$

$\frac{d}{dx} (-2x^3) = 3 \cdot (-2)x^{3-1} = -6x^2$

$\frac{d}{dx} (5x^2) = 2 \cdot 5x^{2-1} = 10x$

$\frac{d}{dx} (-7x) = 1 \cdot (-7)x^{1-1} = -7$

For the last term, $9$ is a constant, so $\frac{d}{dx} (9) = 0$.

Step 2: Combine the terms.

$f'(x) = 12x^3 - 6x^2 + 10x - 7$

Polynomial Example 2

Differentiate $g(x) = 2x^5 - 3x^4 + 6x^3$.

Step 1: Apply the power rule to each term.

$\frac{d}{dx} (2x^5) = 5 \cdot 2x^{5-1} = 10x^4$

$\frac{d}{dx} (-3x^4) = 4 \cdot (-3)x^{4-1} = -12x^3$

$\frac{d}{dx} (6x^3) = 3 \cdot 6x^{3-1} = 18x^2$

Step 2: Combine the terms.

$g'(x) = 10x^4 - 12x^3 + 18x^2$

How to Use This on the AP Calculus Exam

MCQ

When you see a polynomial, differentiate it in one pass: drop each power by one, multiply by the old exponent and any coefficient, and turn constants into zero. Double-check signs on negative terms, since that is the most common spot to lose a point.

Free Response

Show your differentiation clearly so the structure of your work is visible. If you differentiate $3x^4 - 2x^3$, writing the intermediate step $12x^3 - 6x^2$ makes your reasoning easy to follow. Keep notation consistent, using $f'(x)$ or $\frac{dy}{dx}$ throughout, and avoid dropping parentheses on negative coefficients.

Common Trap

A constant term differentiates to $0$, but a constant coefficient on a variable term stays. For example, in $5x$ the $5$ remains as the derivative, while the standalone $+9$ becomes $0$. Mixing these up changes your whole answer.

Common Misconceptions

• The derivative of a constant is zero, not the constant itself. The derivative of $9$ is $0$, not $9$.
• The constant multiple rule keeps the coefficient. The derivative of $5x^3$ is $15x^2$, not $x^2$. You multiply, you do not drop, the coefficient.
• Sums and differences differentiate term by term, but you cannot do the same with products. The product of two functions needs the product rule, which comes later.
• A constant coefficient on a variable term is not the same as a constant term. In $7x$, the $7$ stays attached to the derivative; only a standalone constant goes to zero.
• The exponent drops by exactly one each time. For $-7x$, the power goes from $1$ to $0$, leaving $-7$, not $0$.

Related AP Calculus Guides

• Unit 2 Overview: Differentiation
• 2.1 Defining Average and Instantaneous Rates of Change at a Point
• 2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions
• 2.2 Defining the Derivative of a Function and Using Derivative Notation
• 2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
• 2.3 Estimating Derivatives of a Function at a Point