Edexcel IAL P3 Further Differentiation: exam questions and answers, notes and worked examples

Updated October 2026 · By Pete Bromfield

P3 further differentiation adds the chain, product and quotient rules and the derivatives of $e^{x}$ , $\ln x$ and the trig functions. Spot the structure first: a function of a function needs the chain rule, a product of two functions the product rule, and one function divided by another the quotient rule.

P3 · Pure Mathematics 3
Calculator allowed
3 worked examples
35 practice questions
Tested in 4 of the last 4 papers

Practise P3 the chain, product and quotient rules →

Notes
Formulas
Worked examples
Mistakes
FAQ

P3 Further Differentiation notes

Standard derivatives

$\frac{d}{d x} e^{k x} = k e^{k x}$ , $\frac{d}{d x} \ln x = \frac{1}{x}$ , $\frac{d}{d x} \sin k x = k \cos k x$ , $\frac{d}{d x} \cos k x = - k \sin k x$ and $\frac{d}{d x} \tan x = \sec^{2} x$ . Note that $\frac{d}{d x} \ln 5 x = \frac{1}{x}$ , because $\ln 5 x = \ln 5 + \ln x$ .

The chain rule

$\frac{d y}{d x} = \frac{d y}{d u} \times \frac{d u}{d x}$ : differentiate the outside, keep the inside, multiply by the inside's derivative. $\frac{d}{d x} (3 x^{2} + 1)^{5} = 30 x (3 x^{2} + 1)^{4}$ .

The product rule

$\frac{d}{d x} (u v) = u \frac{d v}{d x} + v \frac{d u}{d x}$ . $\frac{d}{d x} (x^{2} e^{3 x}) = e^{3 x} (3 x^{2} + 2 x)$ . Factorise the answer: it makes stationary points easy to find.

The quotient rule

$\frac{d}{d x} (\frac{u}{v}) = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v^{2}}$ . $\frac{d}{d x} (\frac{\sin x}{x}) = \frac{x \cos x - \sin x}{x^{2}}$ .

Using dx/dy

$\frac{d y}{d x} = \frac{1}{d x / d y}$ . If $x = y^{3} + y$ , then $\frac{d x}{d y} = 3 y^{2} + 1$ , so $\frac{d y}{d x} = \frac{1}{3 y^{2} + 1}$ .

Key formulas

Standard derivatives Learn it: $(e^{k x})^{'} = k e^{k x}, (\ln x)^{'} = \frac{1}{x}, (\sin k x)^{'} = k \cos k x, (\cos k x)^{'} = - k \sin k x$
Trig derivatives In the formula booklet: $(\tan k x)^{'} = k \sec^{2} k x, (\sec x)^{'} = \sec x \tan x, (\cot x)^{'} = - \cosec^{2} x, (\cosec x)^{'} = - \cosec x \cot x$
Product; chain Learn it: $(u v)^{'} = u v^{'} + v u^{'}; \frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}$
Quotient In the formula booklet: $\frac{d}{d x} (\frac{f}{g}) = \frac{f^{'} g - f g^{'}}{g^{2}}$
Inverse Learn it: $\frac{d x}{d y} = \frac{1}{d y / d x}$

In the formula booklet = printed in Pearson's IAL formulae booklet, so know where to find it. Learn it = not printed there, so learn it by heart. All P3 formulas on one page (free PDF) →

P3 Further Differentiation worked examples

Each mark scheme is checked line by line before it is published here. M marks are for method, A marks for accurate answers that follow from them, B marks for independent results.

Worked example 1 · 5 marks · hard

A curve has the equation $y = e^{- x} \sin x$ . Find the exact x-coordinate of the first positive turning point of the curve.

Mark scheme

M1 for using the product rule to differentiate: $\frac{d y}{d x} = - e^{- x} \sin x + e^{- x} \cos x$ .
M1 for setting the derivative equal to 0 to find turning points: $e^{- x} (\cos x - \sin x) = 0$ .
M1 for dividing by $e^{- x}$ (since $e^{- x} \neq 0$ ) to get $\cos x = \sin x$ .
M1 for dividing by $\cos x$ to obtain $\tan x = 1$ .
A1 for identifying the first positive solution as $x = \frac{π}{4}$ .

5 steps

Worked example 2 · 5 marks · hard

Find the equation of the normal to the curve $y = x^{2} e^{- x}$ at the point where $x = 1$ . Give your answer in the form $y = m x + c$ .

Mark scheme

M1 for finding the y-coordinate at $x = 1$ : $y = e^{- 1}$ .
M1 for applying the product rule to find $\frac{d y}{d x} = 2 x e^{- x} - x^{2} e^{- x}$ .
M1 for evaluating the gradient at $x = 1$ : $m_{t} = 2 e^{- 1} - e^{- 1} = e^{- 1}$ .
M1 for finding the perpendicular gradient of the normal: $m_{n} = - e$ .
A1 for forming the equation $y - e^{- 1} = - e (x - 1)$ and rearranging to $y = - e x + e + e^{- 1}$ .

5 steps

Worked example 3 · 4 marks · hard

Differentiate $y = \frac{e^{2 x}}{x^{2} + 1}$ with respect to $x$ , giving your answer as a single algebraic fraction in its simplest form.

Mark scheme

M1 for correctly identifying $u = e^{2 x}$ and $v = x^{2} + 1$ , and their derivatives $2 e^{2 x}$ and $2 x$ .
M1 for substituting into the quotient rule: $\frac{(x^{2} + 1) 2 e^{2 x} - e^{2 x} (2 x)}{(x^{2} + 1)^{2}}$ .
M1 for factorising out $2 e^{2 x}$ in the numerator.
A1 for the correct simplified expression: $\frac{2 e^{2 x} (x^{2} - x + 1)}{(x^{2} + 1)^{2}}$ .

4 steps

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Common P3 the chain, product and quotient rules mistakes

Forgetting the inner derivative. $\frac{d}{d x} e^{x^{2}} = 2 x e^{x^{2}}$ , not $e^{x^{2}}$ .
The quotient rule's order. The numerator is $v u^{'} - u v^{'}$ ; swapping it changes the sign of the answer.
Product rule as a product of derivatives. $\frac{d}{d x} (u v)$ is not $u^{'} v^{'}$ .
Degrees on the calculator. Calculus with trig functions needs radians. Set the calculator to radians before substituting.

Every P3 mistake from the examiner reports, with the paper it came from →

On your calculator: how to find a numerical derivative and definite integral. A calculator is allowed in every IAL unit.

P3 Further Differentiation FAQ

Is the quotient rule in the IAL formula booklet?

On our IAL formula sheet the quotient rule and the derivatives of tan, sec, cot and cosec are marked In the formula booklet; the chain and product rules and the derivatives of eˣ, ln x, sin and cos are marked Learn it.

Can I use the product rule instead of the quotient rule?

Yes: write u/v as u × v⁻¹ and use the product rule with the chain rule. The answer is the same, but you may need to simplify it to match the form a question asks for.

What are the usual chain rule slips?

Take y = (3x² + 1)⁴, whose derivative is 4(3x² + 1)³ × 6x = 24x(3x² + 1)³. The common wrong answers leave out the × 6x, differentiate 3x² as 3x instead of 6x, keep the power as 4 instead of reducing it to 3, or drop the + 1 inside the bracket. Write down the inner function u = 3x² + 1 and du/dx = 6x before you start.

Is a calculator allowed in IAL P3?

Yes. A calculator is allowed in every Edexcel IAL maths unit, P3 included. Method marks are for the working you write down, so show it.

How long is the IAL P3 exam?

Each IAL maths unit, P3 included, is one written paper of 1 hour 30 minutes worth 75 marks.

How often does the chain, product and quotient rules come up in P3?

In our mapping of the 4 Pearson P3 papers from June 2024 to June 2025, the chain, product and quotient rules was tested in 4 of them (7 questions in all). Any topic can be examined in any series.

Next steps

More P3: Functions · Numerical Methods · Further Integration · Algebraic Methods · Trigonometric Functions · Trigonometric Addition Formulae · Exponentials and Logarithms

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Edexcel IAL P3 Further Differentiation: exam questions and answers, notes and worked examples

Try it first

P3 Further Differentiation notes

Standard derivatives

The chain rule

The product rule

The quotient rule

Using dx/dy

Key formulas

P3 Further Differentiation worked examples

Worked example 1 · 5 marks · hard

Mark scheme

Worked example 2 · 5 marks · hard

Mark scheme

Worked example 3 · 4 marks · hard

Mark scheme

Common P3 the chain, product and quotient rules mistakes

P3 Further Differentiation FAQ

Is the quotient rule in the IAL formula booklet?

Can I use the product rule instead of the quotient rule?

What are the usual chain rule slips?

Is a calculator allowed in IAL P3?

How long is the IAL P3 exam?

How often does the chain, product and quotient rules come up in P3?

Next steps

Your P3 revision kit