Consider the function $f(x) = e^x \sin x$ , for $0 \le x \le \pi$ . Use the product rule to find $f'(x)$ . The curve has a local maximum point in the given domain. By setting $f'(x) = 0$ , find the exact $x$ -coordinate of this maximum.

Let $u = e^x \implies u^{\prime} = e^x$ . Let $v = \sin x \implies v^{\prime} = \cos x$ . (M1) $f^{\prime}(x) = u^{\prime}v + uv^{\prime} = e^x \sin x + e^x \cos x$ . A1 $f^{\prime}(x) = e^x(\sin x + \cos x)$ . Set $e^x(\sin x + \cos x) = 0$ . (M1) Since $e^x > 0$ for all $x$ , we must have $\sin x + \cos x = 0$ . $\sin x = -\cos x \implies \frac{\sin x}{\cos x} = -1 \implies \tan x = -1$ . (M1) In the domain $0 \le x \le \pi$ , $\tan x = -1$ exactly at $x = \frac{3\pi}{4}$ . A1

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IB Maths AA SL · Unit 5: Calculus

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

Updated October 2026 · By Pete Bromfield

Differentiate powers with $n x^{n - 1}$ after rewriting roots and fractions as powers. Then match the structure: a function inside a function needs the chain rule, a product needs the product rule and a quotient needs the quotient rule. All three rules and the standard derivatives are in the formula booklet; spotting which one applies is the skill.

51 questions
Paper 1: 44
Paper 2: 7
3 worked examples

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Differentiation Rules notes

The power rule

$\frac{d}{d x} x^{n} = n x^{n - 1}$ for any real $n$ . Rewrite first: $4 \sqrt{x} = 4 x^{1 / 2}$ , so its derivative is $2 x^{- 1 / 2} = \frac{2}{\sqrt{x}}$ . Differentiate a sum term by term; a constant on its own differentiates to 0.

Standard derivatives

$\frac{d}{d x} \sin x = \cos x$ , $\frac{d}{d x} \cos x = - \sin x$ , $\frac{d}{d x} e^{x} = e^{x}$ and $\frac{d}{d x} \ln x = \frac{1}{x}$ . With a linear inside: $\frac{d}{d x} e^{k x} = k e^{k x}$ and $\frac{d}{d x} \sin k x = k \cos k x$ .

The chain rule

For a function of a function, differentiate the outside (leaving the inside alone) and multiply by the derivative of the inside: $\frac{d}{d x} (3 x^{2} + 1)^{5} = 5 (3 x^{2} + 1)^{4} \times 6 x = 30 x (3 x^{2} + 1)^{4}$ .

The product rule

$\frac{d}{d x} (u v) = u \frac{d v}{d x} + v \frac{d u}{d x}$ . For $x^{2} e^{x}$ : $x^{2} e^{x} + 2 x e^{x} = e^{x} (x^{2} + 2 x)$ . Factorise the answer; the next part usually needs it.

The quotient rule

$\frac{d}{d x} (\frac{u}{v}) = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v^{2}}$ . The order in the numerator matters. For $\frac{x}{x + 1}$ : $\frac{(x + 1) - x}{(x + 1)^{2}} = \frac{1}{(x + 1)^{2}}$ .

The second derivative

Differentiate again to get $f^{″} (x)$ , the rate of change of the gradient. It decides the nature of stationary points and the concavity of the graph; see applications of differentiation.

Key formulas

Power rule In the booklet: $f (x) = x^{n} \Rightarrow f^{'} (x) = n x^{n - 1}$
Standard derivatives In the booklet: $(\sin x)^{'} = \cos x, (\cos x)^{'} = - \sin x, (e^{x})^{'} = e^{x}, (\ln x)^{'} = \frac{1}{x}$
Chain rule In the booklet: $y = g (u), u = f (x) \Rightarrow \frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}$
Product rule In the booklet: $\frac{d}{d x} (u v) = u \frac{d v}{d x} + v \frac{d u}{d x}$
Quotient rule In the booklet: $\frac{d}{d x} (\frac{u}{v}) = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v^{2}}$
Linear inside Learn it: $\frac{d}{d x} e^{k x} = k e^{k x}, \frac{d}{d x} \sin k x = k \cos k x, \frac{d}{d x} \ln (k x) = \frac{1}{x}$

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 AA SL formulas →

Explained, with a worked example: Chain rule · Derivative of sin x · Power rule · Product rule · Quotient rule

Differentiation Rules worked examples

Worked example 1: Polynomial power rule and stationary points · easy

The equation of a curve is $y = \frac{3}{2} x^{2} - 15 x + 2$ . Find an expression for $\frac{d y}{d x}$ , and hence find the exact coordinates of the point on the curve where the gradient of the tangent is zero.

Solution

6 steps · 1 to fill in

Examiner tip: The derivative of any constant is zero — the

+ 2

has no effect on the gradient because a constant doesn't change the rate of change of the curve.

Worked example 2: Quotient rule differentiation · medium

A curve has equation $y = \frac{- 7 x}{x^{3} - 1}$ , for $x \neq 1$ . Use the quotient rule to find a fully simplified expression for $\frac{d y}{d x}$ .

Solution

6 steps

Examiner tip: The formula is

u^{'} v - u v^{'}

, NOT

u v^{'} - u^{'} v

. Memorise: "derivative of the top FIRST".

Worked example 3: Product rule with an embedded chain rule · hard

Let $f (x) = (12 x^{2} - 7) e^{- 2 x}$ . Find $f^{'} (x)$ and show it can be written in the fully factorised form $f^{'} (x) = - 2 e^{- 2 x} (a x^{2} + b x + c)$ , where $a, b, c$ are integers.

Solution

6 steps

Examiner tip: Questions that specify a factorised target form are giving you a huge hint. Do not expand — instead pull the required common factor out of every term.

Differentiation Rules exam-style questions

3 questions from our AA 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 · 4 marks · Paper 2

Consider the function with rational exponents $f (x) = x^{\frac{2}{3}} + \frac{5}{x}$ for $x > 0$ .

Rewrite $f (x)$ so that all terms are in the form $a x^{n}$ . Hence, find $f^{'} (x)$ .
Use your graphic display calculator to evaluate the exact gradient of the curve at $x = 8$ .

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Question 2 · medium · 5 marks · Paper 1

Consider the function $f (x) = e^{x} \sin x$ , for $0 \leq x \leq π$ .

Use the product rule to find $f^{'} (x)$ .
The curve has a local maximum point in the given domain. By setting $f^{'} (x) = 0$ , find the exact $x$ -coordinate of this maximum.

Full worked solution Free, no sign-in

Let $u = e^{x} ⟹ u^{'} = e^{x}$ .
Let $v = \sin x ⟹ v^{'} = \cos x$ . (M1)
$f^{'} (x) = u^{'} v + u v^{'} = e^{x} \sin x + e^{x} \cos x$ . A1
$f^{'} (x) = e^{x} (\sin x + \cos x)$ .
Set $e^{x} (\sin x + \cos x) = 0$ . (M1)
Since $e^{x} > 0$ for all $x$ , we must have $\sin x + \cos x = 0$ .
$\sin x = - \cos x ⟹ \frac{\sin x}{\cos x} = - 1 ⟹ \tan x = - 1$ . (M1)
In the domain $0 \leq x \leq π$ , $\tan x = - 1$ exactly at $x = \frac{3 π}{4}$ . A1

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

Question 3 · hard · 5 marks · Paper 1

A curve has the equation $y = e^{- 3 x} + \ln x$ , for $x > 0$ .

Find $\frac{d y}{d x}$ .
Find the exact gradient of the normal line to the curve at the point where $x = 1$ .

Attempt it and see the mark scheme →

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All 51 differentiation rules questions →

Differentiation Rules on your calculator

A GDC is allowed on Paper 2, not on Paper 1. 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 Rules FAQ

Are the chain, product and quotient rules in the IB formula booklet?

Yes, all three are in the AA SL booklet, with the derivatives of xⁿ, sin x, cos x, eˣ and ln x. Learn the quick results for a linear inside, such as the derivative of e^(kx) being k e^(kx).

How do I know which rule to use?

Look at the structure. One function inside another: chain rule. Two functions multiplied: product rule. One divided by another: quotient rule. Many questions need two of them together.

How many IB Maths AA SL rules of differentiation questions are there?

There are 51 exam-style rules of differentiation questions in the AA SL question bank (Paper 1: 44 · Paper 2: 7), graded 12 easy, 23 medium, 13 hard, 3 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is rules of differentiation on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 44 · Paper 2: 7. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Next steps

More AA SL Unit 5: Definite Integrals & Area · Graphical Behaviour and the Second Derivative · Integration · Kinematics · Limits and Rates of Change

Teaching this topic? Free ready-to-teach lessons (slides and teacher notes): Chain rule · Product & quotient rules · Tangents & normals at a point.

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

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

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Differentiation Rules notes

The power rule

Standard derivatives

The chain rule

The product rule

The quotient rule

The second derivative

Key formulas

Differentiation Rules worked examples

Worked example 1: Polynomial power rule and stationary points · easy

Solution

Worked example 2: Quotient rule differentiation · medium

Solution

Worked example 3: Product rule with an embedded chain rule · hard

Solution

Differentiation Rules exam-style questions

Question 1 · easy · 4 marks · Paper 2

Question 2 · medium · 5 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 5 marks · Paper 1

Differentiation Rules on your calculator

Differentiation Rules FAQ

Next steps

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