Constant Multiple Rule

Definition

The Constant Multiple Rule states that the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. Mathematically, if $c$ is a constant and $f(x)$ is a differentiable function, then $(cf(x))' = c f'(x)$.

5 Must Know Facts For Your Next Test

1. The Constant Multiple Rule can be expressed as $(cf(x))' = c f'(x)$ where $c$ is a constant.
2. This rule simplifies differentiation when functions are scaled by constants.
3. It applies to both polynomial and non-polynomial functions as long as they are differentiable.
4. Combining this rule with other differentiation rules like the Product Rule or Chain Rule can simplify complex derivatives.
5. If $f(x) = x^n$, then applying the Constant Multiple Rule gives $(cx^n)' = c(nx^{n-1})$.

Review Questions

• What does the Constant Multiple Rule state about the derivative of a constant times a function?
• How would you apply the Constant Multiple Rule to differentiate $3x^2$?
• Can you use the Constant Multiple Rule in conjunction with other differentiation rules? Provide an example.

Related terms

Power Rule: A basic rule of differentiation used to find the derivative of a function in the form $f(x) = x^n$, resulting in $f'(x) = nx^{n-1}$.

Product Rule: A rule for finding the derivative of the product of two functions, stated as $(fg)' = f'g + fg'$.

Chain Rule: $A rule for computing the derivative of the composition of two or more functions, expressed as (f(g(x)))' = f'(g(x)) \cdot g'(x).