--- title: "AP Calculus 2.6: Constant, Sum, Difference, and Constant Multiple Rules" description: "Review AP Calculus 2.6, including derivative rules for constants, sums, differences, and constant multiples; combining these rules with the power rule; differentiating polynomials term by term; and avoiding common coefficient and sign mistakes." canonical: "https://fiveable.me/ap-calc/unit-2/derivative-rules-constant-sum-difference-constant-multiple/study-guide/0DS7QaXV5BZFYbNZySlm" type: "study-guide" subject: "AP Calculus AB/BC" unit: "Unit 2 – Fundamentals of Differentiation" lastUpdated: "2026-06-09" --- # AP Calculus 2.6: Constant, Sum, Difference, and Constant Multiple Rules ## Summary Review AP Calculus 2.6, including derivative rules for constants, sums, differences, and constant multiples; combining these rules with the power rule; differentiating polynomials term by term; and avoiding common coefficient and sign mistakes. ## Guide These [derivative rules](/ap-calc/unit-2/derivatives-tangent-cotangent-secant-cosecant-functions/study-guide/wkBAhxiFZ1wV1M5mwLwt "fv-autolink") let you differentiate polynomials fast: the [derivative](/ap-calc/unit-10-infinite-sequences-and-series-bc-only/finding-taylor-polynomial-approximations-functions/study-guide/LszguYzKz0M6GdqTRSr6 "fv-autolink") 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](/ap-calc/key-terms/tangent-line "fv-autolink") slopes to locating [critical points](/ap-calc/key-terms/critical-points "fv-autolink") 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](/ap-calc/unit-7/verifying-solutions-for-differential-equations/study-guide/s2nX7AhIBDxGIWwlL82x "fv-autolink"). - In later units, you will reuse these rules constantly when analyzing [increasing](/ap-calc/unit-5/determining-intervals-on-which-function-is-increasing-or-decreasing/study-guide/Y2fgjyl7H1dKPI2YsB4Y "fv-autolink")/decreasing behavior, concavity, and instantaneous [rates of change](/ap-calc/key-terms/rate-of-change "fv-autolink"). 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](/ap-calc/unit-1/defining-continuity-at-point/study-guide/JbsR9iQfAzCznNOCG6JK "fv-autolink") never changes. - The sum and difference rules let you differentiate each term separately, then add or subtract the results. - The [constant multiple rule](/ap-calc/key-terms/constant-multiple-rule "fv-autolink") 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](/ap-calc/unit-2/applying-power-rule/study-guide/GMr6EEbZezsP1DvqrpEk) 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](/ap-calc/key-terms/slope "fv-autolink") 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](/ap-calc/key-terms/sum-rule "fv-autolink"), 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](/ap-calc/key-terms/difference-rule "fv-autolink"). 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](/ap-calc/unit-2/product-rule/study-guide/qQXYTmpHvjsqAWOVzcEw "fv-autolink"), 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](/ap-calc/unit-2/review/study-guide/HGq8OPntbFvqp1uRuEcj) - [2.1 Defining Average and Instantaneous Rates of Change at a Point](/ap-calc/unit-2/defining-average-instantaneous-rates-change-at-point/study-guide/5ozgLc7Ucg4El3O0fV28) - [2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions](/ap-calc/unit-2/derivatives-tangent-cotangent-secant-cosecant-functions/study-guide/wkBAhxiFZ1wV1M5mwLwt) - [2.2 Defining the Derivative of a Function and Using Derivative Notation](/ap-calc/unit-2/defining-derivative-function-using-derivative-notation/study-guide/j9KaEWbB5OECijykhxCD) - [2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist](/ap-calc/unit-2/determining-when-derivatives-do-do-not-exist/study-guide/Lk6aXtIExtqciduDNGhk) - [2.3 Estimating Derivatives of a Function at a Point](/ap-calc/unit-2/estimating-derivatives-function-at-point/study-guide/EzDxIj2eOHj3Ysb68Ew5) ## Vocabulary - **constant multiple rule**: A derivative rule stating that the derivative of a constant times a function equals the constant times the derivative of the function. - **derivative**: The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. - **difference rule**: A derivative rule stating that the derivative of a difference of functions equals the difference of their individual derivatives. - **polynomial function**: A function composed of terms with non-negative integer exponents and real coefficients. - **power rule**: A derivative rule stating that the derivative of x^n is n·x^(n-1), where n is a constant. - **sum rule**: A derivative rule stating that the derivative of a sum of functions equals the sum of their individual derivatives. ## FAQs ### What derivative rules are in AP Calculus 2.6? Topic 2.6 covers the constant rule, sum rule, difference rule, and constant multiple rule. You use these with the power rule to differentiate polynomial functions term by term. ### What is the derivative of a constant? The derivative of a constant is 0. A constant function does not change, so its graph is horizontal and its slope is zero everywhere. ### What is the sum rule for derivatives? The sum rule says 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). The difference rule works the same way with subtraction. ### What is the constant multiple rule? The constant multiple rule says a constant coefficient stays in front when you differentiate: d/dx[cg(x)] = c g'(x). For example, the derivative of 5x^3 is 15x^2, not x^2. ### How do you differentiate a polynomial term by term? Apply the power rule to each variable term, keep coefficients using the constant multiple rule, add or subtract according to the original signs, and turn standalone constants into 0. ### What is the most common mistake with these derivative rules? The most common mistakes are dropping coefficients, forgetting that standalone constants become 0, or changing signs when using the difference rule. Write each term derivative clearly before combining. ## Structured Data ```json {"@context":"https://schema.org","@type":"FAQPage","inLanguage":"en","mainEntity":[{"@type":"Question","@id":"https://fiveable.me/ap-calc/unit-2/derivative-rules-constant-sum-difference-constant-multiple/study-guide/0DS7QaXV5BZFYbNZySlm#what-derivative-rules-are-in-ap-calculus-26","name":"What derivative rules are in AP Calculus 2.6?","acceptedAnswer":{"@type":"Answer","text":"Topic 2.6 covers the constant rule, sum rule, difference rule, and constant multiple rule. You use these with the power rule to differentiate polynomial functions term by term."}},{"@type":"Question","@id":"https://fiveable.me/ap-calc/unit-2/derivative-rules-constant-sum-difference-constant-multiple/study-guide/0DS7QaXV5BZFYbNZySlm#what-is-the-derivative-of-a-constant","name":"What is the derivative of a constant?","acceptedAnswer":{"@type":"Answer","text":"The derivative of a constant is 0. A constant function does not change, so its graph is horizontal and its slope is zero everywhere."}},{"@type":"Question","@id":"https://fiveable.me/ap-calc/unit-2/derivative-rules-constant-sum-difference-constant-multiple/study-guide/0DS7QaXV5BZFYbNZySlm#what-is-the-sum-rule-for-derivatives","name":"What is the sum rule for derivatives?","acceptedAnswer":{"@type":"Answer","text":"The sum rule says 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). The difference rule works the same way with subtraction."}},{"@type":"Question","@id":"https://fiveable.me/ap-calc/unit-2/derivative-rules-constant-sum-difference-constant-multiple/study-guide/0DS7QaXV5BZFYbNZySlm#what-is-the-constant-multiple-rule","name":"What is the constant multiple rule?","acceptedAnswer":{"@type":"Answer","text":"The constant multiple rule says a constant coefficient stays in front when you differentiate: d/dx[cg(x)] = c g'(x). For example, the derivative of 5x^3 is 15x^2, not x^2."}},{"@type":"Question","@id":"https://fiveable.me/ap-calc/unit-2/derivative-rules-constant-sum-difference-constant-multiple/study-guide/0DS7QaXV5BZFYbNZySlm#how-do-you-differentiate-a-polynomial-term-by-term","name":"How do you differentiate a polynomial term by term?","acceptedAnswer":{"@type":"Answer","text":"Apply the power rule to each variable term, keep coefficients using the constant multiple rule, add or subtract according to the original signs, and turn standalone constants into 0."}},{"@type":"Question","@id":"https://fiveable.me/ap-calc/unit-2/derivative-rules-constant-sum-difference-constant-multiple/study-guide/0DS7QaXV5BZFYbNZySlm#what-is-the-most-common-mistake-with-these-derivative-rules","name":"What is the most common mistake with these derivative rules?","acceptedAnswer":{"@type":"Answer","text":"The most common mistakes are dropping coefficients, forgetting that standalone constants become 0, or changing signs when using the difference rule. Write each term derivative clearly before combining."}}]} ```