Differentiation of polynomial functions (power rule)

IB Mathematics AI SL· 5.2· 15 min read

1. The Power Rule for Differentiation★★☆☆☆⏱ 5 min

📘 Definition

Power Rule

For $y = a x^{n}$ , $\frac{d y}{d x} = na x^{n - 1}$

A formula that gives the derivative of any term of the form $a x^{n}$ , where $a$ is a constant and $n$ is any real number. Works for positive, negative and fractional exponents.

Example:

For $y = 3 x^{2}$ , the derivative is $6 x$

The power rule simplifies differentiation of power terms, replacing the need to use first principles for every calculation. It is the most widely used rule in introductory calculus for IB AI SL.

📐 Worked Example

Differentiate the function $y = 5 x^{3}$

1
Identify $a$ (coefficient) and $n$ (exponent) from the form $y = a x^{n}$ :
2
Here, $a = 5$ and $n = 3$
3
Apply the power rule: multiply $a$ by $n$ , then subtract 1 from the exponent:
4
$\frac{dy}{dx} = (3 \times 5)x^{3-1} = 15x^2$

Exam tip:

Always remember to subtract 1 from the original exponent — this is the most common mistake students make in exams.

2. Differentiating Full Polynomial Functions★★☆☆☆⏱ 6 min

A polynomial is a sum of individual power terms. The derivative of a sum equals the sum of the derivatives, so you just apply the power rule to each term separately. Constant terms (which can be written as $a x^{0}$ ) always differentiate to zero.

📐 Worked Example

Differentiate the function $f (x) = 4 x^{4} - 2 x^{2} + 7 x - 10$

1
Apply the power rule to each term one by one:
2
$f'(x) = (4 \times 4)x^{4-1} + (-2 \times 2)x^{2-1} + (7 \times 1)x^{1-1} + 0$
3
Simplify, noting that $x^{0} = 1$ :
4
$f'(x) = 16x^3 - 4x + 7$

✓ Quick check

Test your understanding

What is the derivative of $y = 2 x^{- 3}$ ?
- $- 6 x^{- 4}$
- $- \frac{2}{3} x^{- 4}$
- $- 6 x^{- 2}$

3. Finding the Gradient of a Tangent★★★☆☆⏱ 7 min

🚫 No Calculator

A very common exam question asks for the gradient of the tangent to a polynomial curve at a specific point. The value of the derivative at that point equals the gradient of the tangent there. The process is always: differentiate first, then substitute the given $x$ -value.

📐 Worked Example

Find the gradient of the tangent to $y = x^{3} - 4 x + 2$ at $x = 2$

1
First, differentiate the function term by term using the power rule:
2
$\frac{dy}{dx} = 3x^2 - 4$
3
Substitute $x = 2$ into the derivative to get the gradient at this point:
4
$\frac{dy}{dx} \bigg|_{x=2} = 3(2)^2 - 4 = 12 - 4 = 8$

4. Common Pitfalls

Wrong move:

Forgetting to subtract 1 from the exponent, e.g. writing derivative of $3 x^{2}$ as $6 x^{2}$

Why:

The rule requires reducing the exponent by 1 after multiplying by the original exponent, this step is often skipped

Correct move:

Always apply $n \to n - 1$ : derivative of $3 x^{2} = 6 x^{2 - 1} = 6 x$

Wrong move:

Leaving constant terms in the final derivative

Why:

Students forget constants are $a x^{0}$ , so their derivative is zero

Correct move:

All constant terms in the original polynomial will disappear from the derivative

Wrong move:

Sign errors with negative exponents, e.g. writing derivative of $2 x^{- 2}$ as $- 2 x^{- 1}$

Why:

Incorrect multiplication or exponent subtraction for negative values

Correct move:

Derivative of $2 x^{- 2} = (- 2 \times 2) x^{- 2 - 1} = - 4 x^{- 3}$

Wrong move:

Substituting $x$ into the original function instead of the derivative to get gradient

Why:

Confusing the value of the function ( $y$ ) with the value of the gradient ( $\frac{d y}{d x}$ )

Correct move:

Always differentiate first, then substitute $x$ into the derivative, not the original function

5. Quick Reference Cheatsheet

Term Type	Original Form	Derivative
Single power term	$a x^{n}$	$na x^{n - 1}$
Constant term	$c$	$0$
Linear term	$m x + c$	$m$
General polynomial	$\sum_{n = 0}^{k} a_{n} x^{n}$	$\sum_{n = 1}^{k} n a_{n} x^{n - 1}$
Gradient at $x = a$	$y = f (x)$	$f^{'} (a)$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 1
Find derivative of cubic polynomial
2023 · 1
Find gradient of tangent to quadratic

Going deeper

syllabusIB AI SL Official SyllabusFull topic specification

What's Next

Mastering the power rule for polynomials is the foundation for all further calculus topics in IB AI SL. This rule is used in almost every differentiation question you will encounter on the exam, so fluency here saves time on harder problems. Next, you will use this skill to find full equations of tangents and normals, identify stationary points for graph sketching, and solve real-world optimization problems that are common high-mark exam questions.