# Differentiation of polynomial functions (power rule)

> IB Mathematics AI SL · IB AI SL
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u5-differentiation-of-polynomial-functions/

This foundational sub-topic covers the power rule for differentiating polynomial functions, the core skill for all calculus in IB AI SL. You will learn to find derivatives and calculate gradients of tangents to polynomial curves.

**Prerequisites:** [Polynomial functions and notation](https://www.owlsprep.com/study/ib-math-ai-sl-polynomial-functions/); [Introduction to differentiation as gradient](https://www.owlsprep.com/study/ib-math-ai-sl-u5-introduction-to-differentiation/)

## Learning objectives

- Recall and apply the power rule for differentiation of polynomial terms
- Differentiate any full polynomial function by applying the rule to each term
- Calculate the gradient of a tangent to a polynomial curve at a given point
- Identify and avoid common errors when applying the power rule

## The Power Rule for Differentiation

**Power Rule** — A formula that gives the derivative of any term of the form $ax^n$, where $a$ is a constant and $n$ is any real number. Works for positive, negative and fractional exponents.

*Notation:* For $y = ax^n$, $\frac{dy}{dx} = nax^{n-1}$

*Example:* For $y = 3x^2$, the derivative is $6x$

The power rule gives a fast, direct way to differentiate power terms. It is the most widely used rule in introductory calculus for IB AI SL.

**Worked example:** Differentiate the function $y = 5x^3$

1. Identify $a$ (coefficient) and $n$ (exponent) from the form $y = ax^n$:
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.

## Differentiating Full Polynomial Functions

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 $ax^0$) always differentiate to zero.

**Worked example:** Differentiate the function $f(x) = 4x^4 - 2x^2 + 7x - 10$

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. $$f'(x) = 16x^3 - 4x + 7$$

> **info**
>
> The derivative of a linear term $mx$ is always just the constant $m$, because $1 \times m x^0 = m$.

**Check your understanding**

Test your understanding

1. What is the derivative of $y = 2x^{-3}$?

   - $-6x^{-4}$
   - $-\frac{2}{3}x^{-4}$
   - $-6x^{-2}$

   *Answer:* $-6x^{-4}$

   *Why:* Correct: $a=2$, $n=-3$, so $2 \times (-3) = -6$ and exponent becomes $-3 -1 = -4$.

## Finding the Gradient of a Tangent

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 $y = x^3 - 4x + 2$ 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. $$\frac{dy}{dx} \bigg|_{x=2} = 3(2)^2 - 4 = 12 - 4 = 8$$

**Exam command terms**

- **Find the gradient** — Differentiate, substitute the given $x$ and state the value *(Answer: gradient = 8)*

- **Differentiate the function** — Simplify the derivative fully, leave it in terms of $x$

*Calculator:* forbidden

## Common pitfalls

- **Wrong:** Forgetting to subtract 1 from the exponent, e.g. writing derivative of $3x^2$ as $6x^2$
  - Why it fails: The rule requires reducing the exponent by 1 after multiplying by the original exponent, this step is often skipped
  - Correct: Always apply $n \rightarrow n-1$: derivative of $3x^2 = 6x^{2-1} = 6x$
- **Wrong:** Leaving constant terms in the final derivative
  - Why it fails: Students forget constants are $ax^0$, so their derivative is zero
  - Correct: All constant terms in the original polynomial will disappear from the derivative
- **Wrong:** Sign errors with negative exponents, e.g. writing derivative of $2x^{-2}$ as $-2x^{-1}$
  - Why it fails: Incorrect multiplication or exponent subtraction for negative values
  - Correct: Derivative of $2x^{-2} = (-2 \times 2)x^{-2-1} = -4x^{-3}$
- **Wrong:** Substituting $x$ into the original function instead of the derivative to get gradient
  - Why it fails: Confusing the value of the function ($y$) with the value of the gradient ($\frac{dy}{dx}$)
  - Correct: Always differentiate first, then substitute $x$ into the derivative, not the original function

## Cheatsheet

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

## 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.

- [Tangents, gradients, and stationary points](https://www.owlsprep.com/study/ib-math-ai-sl-u5-tangents-gradients-and-stationary-points/)
- [Optimisation problems with differentiation](https://www.owlsprep.com/study/ib-math-ai-sl-u5-optimisation-problems-with-differentiation/)
- [Indefinite integration of polynomials](https://www.owlsprep.com/study/ib-math-ai-sl-u5-indefinite-integration-of-polynomials/)

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