The constant multiple rule states that the derivative of a constant times a function equals the constant times the derivative of the function: d/dx[c·f(x)] = c·f'(x). In AP Calculus, it lets you pull constants out front and differentiate only the function part.
The constant multiple rule is one of the basic differentiation rules. It says that if you have a constant multiplied by a function, the constant just rides along while you differentiate. In symbols, d/dx[c·f(x)] = c·f'(x). So the derivative of 5x³ is 5·(3x²) = 15x², and the derivative of -2sin(x) is -2cos(x). The constant never changes; only the function gets differentiated.
Why does this work? Derivatives measure rate of change, and multiplying a function by 5 makes it change 5 times as fast at every point. The rule comes straight from the limit definition of the derivative, where the constant factors out of the limit. On the AP exam you'll almost never apply it alone. It works as a teammate, combining with the power rule, sum/difference rule, and later the chain rule to let you differentiate full polynomials and messier expressions term by term.
This rule lives in Unit 2 (Differentiation: Definition and Fundamental Properties), where you build the toolbox of derivative rules that the entire rest of the course depends on. Together with the power rule and the sum/difference rule, it's what makes derivatives of polynomials fast. Instead of grinding through the limit definition for 4x³ - 7x² + 2x, you differentiate each term in seconds. Every later skill, finding instantaneous rates of change, locating critical points in Unit 5, setting up related rates in Unit 4, assumes you can apply this rule automatically. If it's not second nature, everything downstream slows down. The good news is that it's the most forgiving rule in calculus, because linearity (constants pull out, terms split apart) is exactly how you'd hope derivatives behave.
Keep studying AP Calculus Unit f0gqO6h7yKA5SxWY
Sum/Difference Rule (Unit 2)
These two rules are a package deal. Together they make differentiation 'linear,' meaning you can split a function into terms and pull constants out of each one. That's the entire reason you can differentiate any polynomial term by term.
Chain Rule (Unit 3)
A classic trap is treating something like sin(3x) as a constant multiple problem. It's not. The 3 is inside the function, so you need the chain rule, which gives 3cos(3x). The constant multiple rule only applies when the constant multiplies the whole function from the outside, like 3sin(x).
Quotient Rule (Unit 2)
Dividing by a constant is secretly a constant multiple problem. The derivative of f(x)/5 doesn't need the quotient rule; rewrite it as (1/5)·f(x) and pull the 1/5 out. Saving the quotient rule for actual variable denominators saves time and errors.
Instantaneous Rate of Change (Unit 2)
The rule has a physical meaning. If position is scaled by a constant, velocity scales by the same constant. Doubling a function doubles how fast it changes at every single point, which is exactly what c·f'(x) says.
You won't see an FRQ that says 'use the constant multiple rule.' Instead, it's baked into nearly every derivative you compute, on both the multiple-choice and free-response sections. MCQs test it inside questions like 'find f'(x) if f(x) = 4x³ - 2x' or in disguised forms like differentiating x²/3 (rewrite as (1/3)x² first). The most common scoring mistakes are differentiating the constant too (the derivative of 5x² is 10x, not 5·2x·something extra) and confusing an outside constant with an inside one, which needs the chain rule instead. On FRQs, a dropped or mangled constant in a derivative usually costs the answer point, so apply this rule cleanly and move on.
The product rule is for two functions multiplied together, like x²·sin(x). The constant multiple rule is for a number times a function, like 5·sin(x). Technically the product rule still works on 5·sin(x), since the derivative of 5 is 0, the extra term vanishes and you get 5cos(x) either way. But using the product rule there wastes time. Quick test: if one factor has no variable in it, it's a constant multiple, so just pull it out front.
The constant multiple rule says d/dx[c·f(x)] = c·f'(x), so a constant factor passes straight through the derivative unchanged.
It only applies when the constant multiplies the entire function from outside; a constant inside the function, like the 3 in sin(3x), triggers the chain rule instead.
Combined with the sum/difference rule and power rule, it lets you differentiate any polynomial term by term without the limit definition.
Division by a constant counts too: rewrite f(x)/5 as (1/5)·f(x) instead of reaching for the quotient rule.
Conceptually, scaling a function by c scales its rate of change by c at every point, which is why the rule works.
You could use the product rule on c·f(x) and get the same answer, but pulling the constant out is faster and less error-prone on the exam.
It's the rule that d/dx[c·f(x)] = c·f'(x). When a constant multiplies a function, you keep the constant and differentiate only the function, so the derivative of 7x⁴ is 7·4x³ = 28x³.
No, that's a different rule. A constant by itself (like f(x) = 7) has derivative 0, but a constant multiplying a function sticks around. The derivative of 7x² is 14x, not 0 and not 2x.
The product rule handles two functions of x multiplied together, like x²·eˣ. The constant multiple rule handles a plain number times a function. The product rule technically still works on a constant times a function, but it's overkill since one of the terms is always zero.
Chain rule. The 3 is inside the function's input, not multiplying the function from outside, so the derivative is 3cos(3x). Compare that with 3sin(x), where the constant multiple rule gives 3cos(x).
Yes. Dividing by 4 is the same as multiplying by 1/4, so rewrite x³/4 as (1/4)x³ and differentiate to get (3/4)x². No quotient rule needed when the denominator is just a number.