Introduction to the Power Rule

Differentiation is the process of finding the rate of change of a function. For polynomials — functions made up of terms like $x^{n}$ — there is a single elegant rule that handles everything: the power rule.

Power Rule

For a function $f (x) = a x^{n}$ , where $a$ is a constant and $n \in Z$ , the derivative is:
$f^{'} (x) = an x^{n - 1}$
In words: multiply the coefficient by the exponent, then reduce the exponent by 1.

This rule is the backbone of polynomial differentiation and will appear constantly throughout your IB DP Mathematics course. Once mastered, it allows you to differentiate almost any polynomial expression quickly and confidently.

Note

Although in this subtopic we focus on integer exponents ( $n \in Z$ ), the power rule actually extends to all real exponents — including fractions and negatives — as you will see in later sections.

Applying the Power Rule — Step by Step

Applying the power rule is a two-step process:

Multiply the existing coefficient by the exponent
Reduce the exponent by 1

It sounds simple — and it really is, once you drill the steps.

Example

Differentiate $f (x) = 3 x^{4}$

Step 1: Multiply the coefficient by the exponent: $3 \times 4 = 12$

Step 2: Reduce the exponent by 1: $4 - 1 = 3$

$f^{'} (x) = 12 x^{3}$

Example

Differentiate $f (x) = - 5 x^{6}$

Step 1: $- 5 \times 6 = - 30$

Step 2: $6 - 1 = 5$

$f^{'} (x) = - 30 x^{5}$

Exam Tip

Write it out as a formula slot: $(coeff \times power) x^{(power - 1)}$ . Running through this mentally for each term will help you avoid careless errors.

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SL 5.3—Differentiating polynomials, n E Z

Introduction to the Power Rule

Applying the Power Rule — Step by Step

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