The difference rule states that the derivative of a difference of functions equals the difference of their derivatives: d/dx[f(x) − g(x)] = f′(x) − g′(x). In AP Calculus it lets you differentiate term by term, and a matching limit property lets you split limits of differences the same way.
The difference rule is the permission slip that lets you take a derivative one term at a time. If h(x) = f(x) − g(x), then h′(x) = f′(x) − g′(x). Subtraction passes right through the derivative. Combine it with the power rule and the constant multiple rule and you can differentiate any polynomial without ever touching the limit definition. That trio is exactly what the CED means when it says sums, differences, and constant multiples of functions can be differentiated using derivative rules (Topic 2.6).
The same idea actually shows up earlier, in Unit 1. One of the limit theorems in Topic 1.5 says the limit of a difference is the difference of the limits. That's not a coincidence. Derivatives are defined as limits, so the difference rule for derivatives is really the difference rule for limits wearing a Unit 2 costume. When you split lim(x→3) (4x² − 7x) into lim 4x² minus lim 7x, you're using the same structural move you'll use later to differentiate 4x² − 7x term by term.
The difference rule lives in Topic 2.6 (Derivative Rules) and directly supports learning objective AP Calc 2.6.A, calculating derivatives of familiar functions. The essential knowledge there is explicit that the power rule combined with sum, difference, and constant multiple properties is how you find derivatives of polynomial functions. Its limit version supports AP Calc 1.5.A in Unit 1, where EK LIM-1.D.2 says limits of differences can be found using limit theorems. Practically, this rule is invisible glue. Almost every derivative you compute for the rest of the course, from related rates in Unit 4 to optimization in Unit 5 to integration checks in Unit 6, starts with breaking an expression into terms. If you can't split a polynomial apart correctly, every later skill wobbles.
Keep studying AP Calculus Unit 2
Visual cheatsheet
view galleryPower Rule (Unit 2)
The difference rule tells you that you're allowed to differentiate term by term, and the power rule tells you what each term becomes. Together they turn d/dx(x³ − 5x) into 3x² − 5 in one clean pass.
Constant Multiple Rule (Unit 2)
These rules almost always travel together. For h(x) = 5f(x) − 3g(x), the difference rule splits the subtraction and the constant multiple rule pulls out the 5 and 3, giving h′(x) = 5f′(x) − 3g′(x).
Limit Properties (Unit 1)
Topic 1.5 has the exact same rule for limits. The limit of a difference is the difference of the limits. Since a derivative is just a special limit, the derivative version is the limit version inherited one level up.
Chain Rule (Unit 3)
The difference rule works only when functions are subtracted, not composed. Once subtraction shows up inside another function, like (x² − 1)⁵, you need the chain rule on the outside and the difference rule on the inside.
You'll never see an MCQ that says "state the difference rule." Instead, the rule is baked into nearly every derivative computation. Two common formats show up. First, abstract function questions like h(x) = 5f(x) − 3g(x) where you're given f′(2) = 4 and g′(2) = −1 and asked for h′(2). The whole question is testing whether you know h′(2) = 5f′(2) − 3g′(2) = 20 − (−3) = 23, with no formula for f or g ever given. Second, Unit 1 questions ask which limit property justifies splitting something like lim(x→3) (4x² − 7x) into two limits. On FRQs, the difference rule won't be named, but you use it silently every time you differentiate a polynomial to find critical points, slopes, or rates. Watch the sign on the second term; dropping a negative is the classic point-loser here.
Similar names, totally different jobs. The difference quotient is [f(x+h) − f(x)]/h, the expression whose limit defines the derivative in the first place. The difference rule is a shortcut property that says d/dx[f − g] = f′ − g′. One builds derivatives from scratch; the other lets you split a derivative across subtraction without going back to scratch.
The difference rule says the derivative of f(x) − g(x) is f′(x) − g′(x), so you can differentiate term by term.
The same property exists for limits in Topic 1.5, where the limit of a difference equals the difference of the limits.
Combining the difference rule with the power rule and constant multiple rule lets you differentiate any polynomial quickly.
The rule works for abstract functions too. If h(x) = 5f(x) − 3g(x) and you know f′ and g′ at a point, you can find h′ there without knowing the formulas.
The difference rule only applies to subtraction of functions. Quotients need the quotient rule and compositions need the chain rule.
The most common error is a sign mistake on the subtracted term, especially when that term's derivative is already negative.
The difference rule states that d/dx[f(x) − g(x)] = f′(x) − g′(x). The derivative of a subtraction is the subtraction of the derivatives, which is what lets you differentiate polynomials one term at a time.
No. The difference quotient, [f(x+h) − f(x)]/h, is the limit expression that defines a derivative. The difference rule is a shortcut property for splitting a derivative across subtraction. They share a word, not a job.
Yes. EK LIM-1.D.2 in Topic 1.5 says limits of differences can be found using limit theorems, so lim(x→3)(4x² − 7x) = lim 4x² − lim 7x. The derivative version in Unit 2 is built directly on this limit property.
No. The difference rule only handles subtraction. Division requires the quotient rule and composition requires the chain rule. Treating d/dx[f/g] as f′/g′ is one of the most common wrong-answer traps on multiple choice.
Apply the constant multiple and difference rules: h′(x) = 5f′(x) − 3g′(x). So if f′(2) = 4 and g′(2) = −1, then h′(2) = 5(4) − 3(−1) = 23. Watch that double negative.
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