Mathwords logoMathwords

Constant Multiple Rule — Definition, Formula & Examples

The Constant Multiple Rule says that when you take the derivative of a constant multiplied by a function, you can pull the constant out front and just differentiate the function. In short, constants pass through the derivative operator unchanged.

If cc is a real constant and f(x)f(x) is a differentiable function, then ddx[c⋅f(x)]=c⋅f′(x)\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x). The rule follows directly from the limit definition of the derivative, since the constant cc factors out of the limit.

Key Formula

ddx[c⋅f(x)]=c⋅f′(x)\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)
Where:
  • cc = A real-number constant
  • f(x)f(x) = A differentiable function of x
  • f′(x)f'(x) = The derivative of f with respect to x

How It Works

Identify the constant factor in front of the function you need to differentiate. Keep that constant in place, then differentiate only the variable part using whatever rule applies (power rule, trig rule, etc.). Multiply the constant by the result. This rule works because differentiation is a linear operation — scaling a function by a constant scales its rate of change by the same factor.

Worked Example

Problem: Find the derivative of y=7x4y = 7x^4.
Identify the constant: The constant multiple is 7 and the function is x4x^4.
c=7,f(x)=x4c = 7, \quad f(x) = x^4
Differentiate the function: Apply the power rule to x4x^4.
f′(x)=4x3f'(x) = 4x^3
Multiply by the constant: Keep the 7 in front and multiply.
y′=7⋅4x3=28x3y' = 7 \cdot 4x^3 = 28x^3
Answer: y′=28x3y' = 28x^3

Why It Matters

Nearly every derivative you compute in AP Calculus involves pulling out a constant before applying another rule. Mastering this small step prevents arithmetic errors and speeds up problems on the AP exam, especially when combined with the chain rule or product rule on more complex expressions.

Common Mistakes

Mistake: Trying to differentiate the constant itself (treating cc as a variable and applying the power rule to it).
Correction: A constant has a derivative of zero on its own. In the constant multiple rule, cc is not being differentiated — it simply stays as a multiplier in front of f′(x)f'(x).

Related Terms