Consider the function $h(x) = \frac{3x^4 - 2x^2}{x}$ for $x \neq 0$ . Rewrite $h(x)$ as a polynomial by dividing each term in the numerator by $x$ . Find $h'(x)$ . Determine the value of $x$ for which the gradient of $h(x)$ is $23$ .

Divide each term in the numerator by $x$ : $h(x) = \frac{3x^4}{x} - \frac{2x^2}{x}$ . $h(x) = 3x^3 - 2x$ . A1 Differentiate the polynomial: $h'(x) = 9x^2 - 2$ . A1 Set the derivative equal to 23: $9x^2 - 2 = 23$ . (M1) $9x^2 = 25 \implies x^2 = \frac{25}{9}$ . $x = \pm \frac{5}{3}$ . A1

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IB Maths AI SL · Unit 5: Calculus

Differentiation (IB Maths AI SL): notes, exam questions and worked examples

Q: Do I need the chain rule or product rule in AI SL?

No. AI SL differentiates terms of the form axⁿ (n an integer). The chain, product and quotient rules are in AA, and in AI HL.

Q: What is the difference between f(2) and f′(2)?

f(2) is the y-value of the curve at x = 2. f′(2) is the gradient of the curve there. Tangent and normal questions need both.

Q: How many IB Maths AI SL differentiation rules questions are there?

There are 28 exam-style differentiation rules questions in the AI SL question bank (Paper 1: 23 · Paper 2: 5), graded 8 easy, 13 medium, 4 hard, 3 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Q: Is differentiation rules on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 23 · Paper 2: 5. Practise with your GDC — AI papers expect calculator methods throughout.

Updated October 2026 · By Pete Bromfield

The derivative $f^{'} (x)$ is the gradient of the curve and the instantaneous rate of change. In AI SL you differentiate terms $a x^{n}$ (including negative $n$ ) with $n a x^{n - 1}$ , then use $f^{'} (x)$ for the gradient at a point, tangents and normals, where a function increases or decreases, and maximum or minimum values.

28 questions
Paper 1: 23
Paper 2: 5
3 worked examples

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Notes
Formulas
Worked examples
Questions
FAQ

Differentiation notes

Differentiating ax^n

Multiply by the power and reduce the power by one: $\frac{d}{d x} (a x^{n}) = a n x^{n - 1}$ . Differentiate a sum term by term; a constant term differentiates to 0. Rewrite fractions as powers first: $\frac{3}{x^{2}} = 3 x^{- 2}$ , whose derivative is $- 6 x^{- 3}$ .

The gradient at a point

Substitute into $f^{'} (x)$ , not $f (x)$ . For $f (x) = x^{3} - 4 x$ , $f^{'} (x) = 3 x^{2} - 4$ and the gradient at $x = 2$ is $f^{'} (2) = 8$ . To find where the gradient has a given value, solve $f^{'} (x) = k$ .

Tangents and normals

The tangent at $x = a$ has gradient $f^{'} (a)$ and passes through $(a, f (a))$ ; the normal has gradient $- \frac{1}{f^{'} (a)}$ . For the curve above at $x = 2$ : $f (2) = 0$ , so the tangent is $y = 8 x - 16$ .

Increasing and decreasing

$f$ is increasing where $f^{'} (x) > 0$ and decreasing where $f^{'} (x) < 0$ . Sketch $y = f^{'} (x)$ on the GDC to see where it is above or below the axis.

Maximum and minimum points

At a turning point $f^{'} (x) = 0$ . Solve it (by hand or on the GDC) and check the sign of $f^{'}$ either side: $+$ then $-$ is a maximum, $-$ then $+$ a minimum. Optimisation questions build the function from the context first, then do exactly this.

Key formulas

Derivative of $x^{n}$ In the booklet: $f (x) = x^{n} \Rightarrow f^{'} (x) = n x^{n - 1}$
Constant multiples, sums Learn it: $f (x) = a x^{n} + b x^{m} \Rightarrow f^{'} (x) = a n x^{n - 1} + b m x^{m - 1}$
Gradient of tangent at $x = a$ Learn it: $f^{'} (a); tangent y - f (a) = f^{'} (a) (x - a)$
Gradient of normal Learn it: $- \frac{1}{f^{'} (a)}$
Increasing / decreasing Learn it: $f^{'} (x) > 0; f^{'} (x) < 0$
Stationary points; optimisation Learn it: $f^{'} (x) = 0; check the sign of f^{'} either side$

The IB formula booklet is provided in every exam paper. In the booklet = printed there, so know where to find it; Learn it = not in the booklet, so learn it by heart. All AI SL formulas →

Explained, with a worked example: Power rule

Differentiation worked examples

Worked example 1: Power rule and evaluating $f^{'} (x)$ · easy

$f (x) = 4 x^{3} - 5 x^{2} + 2 x - 7$ . Find $f^{'} (x)$ and the exact gradient at $x = 2$ .

Solution

4 steps

Examiner tip: The derivative of a constant (here

- 7

) is zero — a horizontal line has no gradient. The derivative of

2 x

is simply

2

Worked example 2: Finding $x$ given a specific gradient · medium

$g (x) = 2 x^{3} - 3 x^{2} - 12 x$ . Find the exact values of $x$ where the gradient equals $24$ .

Solution

6 steps

Examiner tip: "Gradient equals

k

" means set the DERIVATIVE equal to

k

, not the original function.

Worked example 3: Finding constants from a point and a gradient · hard

Curve $y = a x^{2} + b x$ passes through $(1, 5)$ and has gradient $14$ at $x = 2$ . Find $a$ and $b$ exactly.

Solution

6 steps · 1 to fill in

Examiner tip: A point tells you about

f (x)

; a gradient tells you about

f^{'} (x)

. Keep the two substitutions separate when building simultaneous equations.

Differentiation exam-style questions

3 questions from our AI SL bank, written in the style of IB papers, easiest first. One has its full worked solution open here. Open the worked answer, or check your answer in the practice area.

Question 1 · easy · 3 marks · Paper 1

Consider the polynomial function $f (x) = 4 x^{3} - 5 x^{2} + 2 x - 7$ .

Find the derivative function, $f^{'} (x)$ .
Calculate the exact value of $f^{'} (2)$ .

Check your answer in practice →

Question 2 · easy · 4 marks · Paper 1

Let $y = - 0.5 x^{4} + 3 x^{3} - x$ .

Use the numerical derivative feature (d/dx) on your graphic display calculator to find the gradient of the curve at $x = 3$ .
Interpret the geometric meaning of your answer to part (a).

Check your answer in practice →

Question 3 · medium · 4 marks · Paper 1

Consider the function $h (x) = \frac{3 x^{4} - 2 x^{2}}{x}$ for $x \neq 0$ .

Rewrite $h (x)$ as a polynomial by dividing each term in the numerator by $x$ .
Find $h^{'} (x)$ .
Determine the value of $x$ for which the gradient of $h (x)$ is $23$ .

Full worked solution Free, no sign-in

Divide each term in the numerator by $x$ :
$h (x) = \frac{3 x^{4}}{x} - \frac{2 x^{2}}{x}$ .
$h (x) = 3 x^{3} - 2 x$ . A1
Differentiate the polynomial:
$h^{'} (x) = 9 x^{2} - 2$ . A1
Set the derivative equal to 23: $9 x^{2} - 2 = 23$ . (M1)
$9 x^{2} = 25 ⟹ x^{2} = \frac{25}{9}$ .
$x = \pm \frac{5}{3}$ . A1

Every question here has a full worked solution. See them all →

Free account — sign in to download Topic PDF: 7 questions with worked answers · 4 pages · 221 KB

All 28 differentiation questions →

Where students lose marks on differentiation

Examiner's noteCalculus: not differentiating, f(a) for f′(a), and the derivative as a rate

Where marks are lost: Candidates used a normal's gradient as the tangent's, never differentiated, found f(1) where f′(1) was needed, stopped at f′(1) without writing the tangent's equation, ignored an instruction to use their derivative, stopped at the x-coordinate of a minimum, described a derivative as a height or an average rate, and left out the constant of integration.

Examiners flagged this in 3 exam series (May 2024, November 2024, November 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →

Differentiation on your calculator

A GDC is allowed on every AI paper. Keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire, with a checked worked example:

Numerical derivative and integral on your calculator

Differentiation FAQ

Do I need the chain rule or product rule in AI SL?

No. AI SL differentiates terms of the form axⁿ (n an integer). The chain, product and quotient rules are in AA, and in AI HL.

What is the difference between f(2) and f′(2)?

f(2) is the y-value of the curve at x = 2. f′(2) is the gradient of the curve there. Tangent and normal questions need both.

How many IB Maths AI SL differentiation rules questions are there?

There are 28 exam-style differentiation rules questions in the AI SL question bank (Paper 1: 23 · Paper 2: 5), graded 8 easy, 13 medium, 4 hard, 3 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is differentiation rules on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 23 · Paper 2: 5. Practise with your GDC — AI papers expect calculator methods throughout.

Next steps

More AI SL Unit 5: Limits, Derivatives & Rates of Change · Optimisation · Tangents & Normals

Teaching this topic? Free ready-to-teach lesson (slides and teacher notes): Power-rule differentiation.

Keep revising AI SL: AI SL revision notes · IB Maths AI SL past papers · AI SL common exam mistakes · AI SL syllabus checklist · AI SL calculus knowledge organiser

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← All IB Maths AI SL topics

Differentiation (IB Maths AI SL): notes, exam questions and worked examples

Try it first

Differentiation notes

Differentiating ax^n

The gradient at a point

Tangents and normals

Increasing and decreasing

Maximum and minimum points

Key formulas

Differentiation worked examples

Worked example 1: Power rule and evaluating f′(x) · easy

Solution

Worked example 2: Finding x given a specific gradient · medium

Solution

Worked example 3: Finding constants from a point and a gradient · hard

Solution

Differentiation exam-style questions

Question 1 · easy · 3 marks · Paper 1

Question 2 · easy · 4 marks · Paper 1

Question 3 · medium · 4 marks · Paper 1

Full worked solution Free, no sign-in

Where students lose marks on differentiation

Differentiation on your calculator

Differentiation FAQ

Next steps

What else is free here

Worked example 1: Power rule and evaluating $f^{'} (x)$ · easy

Worked example 2: Finding $x$ given a specific gradient · medium