Edexcel IAL P1 Differentiation: exam questions and answers, notes and worked examples

Updated October 2026 · By Pete Bromfield

In P1 you differentiate sums of powers of $x$ , find gradients, and write the equations of tangents and normals; you also meet the derivative as the limit of a chord's gradient. Rewrite roots and fractions as powers of $x$ before you differentiate. The normal is perpendicular to the tangent, so its gradient is $- \frac{1}{m}$ .

P1 · Pure Mathematics 1
Calculator allowed
3 worked examples
32 practice questions
Tested in 4 of the last 4 papers

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Notes
Formulas
Worked examples
Mistakes
FAQ

P1 Differentiation notes

Differentiation from first principles

The gradient of the tangent is the limit of the gradient of a chord: $f^{'} (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$ . For $f (x) = x^{3}$ , $\frac{(x + h)^{3} - x^{3}}{h} = 3 x^{2} + 3 x h + h^{2}$ , which tends to $3 x^{2}$ as $h \to 0$ . Show the expansion, the cancelling of $h$ and the limit.

Differentiating powers of x

$\frac{d}{d x} x^{n} = n x^{n - 1}$ for any rational $n$ , one term at a time. First write every term as a power: $y = 4 \sqrt{x} - \frac{3}{x^{2}} = 4 x^{\frac{1}{2}} - 3 x^{- 2}$ , so $\frac{d y}{d x} = 2 x^{- \frac{1}{2}} + 6 x^{- 3}$ . Expand brackets and split fractions before differentiating; there is no product or quotient rule in P1.

Gradients, tangents and normals

Substitute the $x$ -value into $\frac{d y}{d x}$ for the gradient $m$ , find $y$ from the curve's equation, then use $y - y_{1} = m (x - x_{1})$ . On $y = x^{2} - 5 x + 4$ at $(1, 0)$ the gradient is $- 3$ , so the tangent is $y = - 3 x + 3$ and the normal, with gradient $\frac{1}{3}$ , is $y = \frac{1}{3} (x - 1)$ .

The second derivative

Differentiate twice for $\frac{d^{2} y}{d x^{2}}$ , the rate of change of the gradient. For $y = x^{3} - 6 x^{2}$ : $\frac{d y}{d x} = 3 x^{2} - 12 x$ and $\frac{d^{2} y}{d x^{2}} = 6 x - 12$ . Using it to classify stationary points comes in P2.

Key formulas

Differentiate / integrate $x^{n}$ Learn it: $\frac{d}{d x} x^{n} = n x^{n - 1}, \int x^{n} d x = \frac{x^{n + 1}}{n + 1} + c (n \neq - 1)$
Tangent; normal Learn it: $y - y_{1} = m (x - x_{1}), m_{n} = - \frac{1}{m}$

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 P1 formulas on one page (free PDF) →

P1 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 · 6 marks · hard

The tangents to the curve $y = x^{2} - 4 x$ at the points where $x = 1$ and $x = 3$ intersect at a point $P$ . Find the coordinates of $P$ .

Mark scheme

M1 for finding the y-coordinates: at $x = 1, y = - 3$ and at $x = 3, y = - 3$ .
M1 for differentiating $\frac{d y}{d x} = 2 x - 4$ to find the gradients: $m_{1} = - 2$ (at $x = 1$ ) and $m_{3} = 2$ (at $x = 3$ ).
M1 for finding the equation of the first tangent: $y - (- 3) = - 2 (x - 1) ⟹ y = - 2 x - 1$ .
M1 for finding the equation of the second tangent: $y - (- 3) = 2 (x - 3) ⟹ y = 2 x - 9$ .
M1 for equating the two tangent equations: $- 2 x - 1 = 2 x - 9 ⟹ 4 x = 8 ⟹ x = 2$ .
A1 for finding the corresponding y-coordinate $y = - 5$ , giving $P (2, - 5)$ .

6 steps · 1 to fill in

Worked example 2 · 6 marks · hard

The normal to the curve $y = \sqrt{x}$ at the point where $x = 4$ cuts the x-axis at $A$ and the y-axis at $B$ . Calculate the exact area of the triangle $A O B$ , where $O$ is the origin.

Mark scheme

M1 for finding the y-coordinate at $x = 4$ : $y = 2$ .
M1 for finding $\frac{d y}{d x} = \frac{1}{2} x^{- \frac{1}{2}}$ and evaluating at $x = 4$ to get gradient $m_{t} = \frac{1}{4}$ .
M1 for finding the normal gradient $m_{n} = - 4$ and its equation: $y - 2 = - 4 (x - 4) ⟹ y = - 4 x + 18$ .
M1 for finding the axis intercepts of the normal: at $y = 0, 4 x = 18 ⟹ x = 4.5$ (point $A$ ) and at $x = 0, y = 18$ (point $B$ ).
M1 for using the area of a triangle formula $\frac{1}{2} \times base \times height = \frac{1}{2} \times 4.5 \times 18$ .
A1 for the final area of $40.5$ square units.

6 steps · 1 to fill in

Worked example 3 · 5 marks · hard

Find the equation of the normal to the curve $y = x^{3} - 2 x$ at the point where $x = 2$ .

Mark scheme

M1 for finding the y-coordinate at $x = 2$ : $y = 2^{3} - 2 (2) = 4$ . Point is $(2, 4)$ .
M1 for differentiating to find $\frac{d y}{d x} = 3 x^{2} - 2$ .
M1 for finding the tangent gradient $m_{t} = 3 (4) - 2 = 10$ , and hence the normal gradient $m_{n} = - \frac{1}{10}$ .
M1 for setting up the normal equation: $y - 4 = - \frac{1}{10} (x - 2)$ .
A1 for the correct equation, e.g. $10 y + x = 42$ or $y = - 0.1 x + 4.2$ .

5 steps · 1 to fill in

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Common P1 differentiation mistakes

Differentiating before rewriting. $\frac{1}{x^{2}}$ is $x^{- 2}$ , so its derivative is $- 2 x^{- 3}$ , not $\frac{1}{2 x}$ . Rewrite every root and fraction as a power first.
Leaving the gradient as a function. The tangent's gradient is a number: substitute the $x$ -value into $\frac{d y}{d x}$ before you write the line.
The wrong normal gradient. The normal's gradient is $- \frac{1}{m}$ , not $- m$ or $\frac{1}{m}$ .
Taking y from the derivative. The point on the curve comes from the curve's own equation, not from $\frac{d y}{d x}$ .

Every P1 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.

P1 Differentiation FAQ

Is differentiation from first principles in IAL P1?

Yes. P1 includes the gradient of a tangent as the limit of the gradient of a chord, so be ready to show the expansion, the cancelling of h and the limit for simple powers of x.

What is the difference between a tangent and a normal?

The tangent touches the curve at the point and has the curve's gradient there. The normal goes through the same point at right angles to the tangent, so its gradient is −1 divided by the tangent's gradient.

Is a calculator allowed in IAL P1?

Yes. A calculator is allowed in every Edexcel IAL maths unit, P1 included. Method marks are for the working you write down, so show it.

How long is the IAL P1 exam?

Each IAL maths unit, P1 included, is one written paper of 1 hour 30 minutes worth 75 marks.

How often does differentiation come up in P1?

In our mapping of the 4 Pearson P1 papers from June 2024 to June 2025, differentiation was tested in 4 of them (7 questions in all). Any topic can be examined in any series.

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Other units: IAL P2 · IAL P3 · IAL P4 · IAL S1 · IAL M1

Edexcel IAL P1 Differentiation: exam questions and answers, notes and worked examples

Try it first

P1 Differentiation notes

Differentiation from first principles

Differentiating powers of x

Gradients, tangents and normals

The second derivative

Key formulas

P1 Differentiation worked examples

Worked example 1 · 6 marks · hard

Mark scheme

Worked example 2 · 6 marks · hard

Mark scheme

Worked example 3 · 5 marks · hard

Mark scheme

Common P1 differentiation mistakes

P1 Differentiation FAQ

Is differentiation from first principles in IAL P1?

What is the difference between a tangent and a normal?

Is a calculator allowed in IAL P1?

How long is the IAL P1 exam?

How often does differentiation come up in P1?

Next steps

Your P1 revision kit