AQA A-Level Maths: Calculus and Coordinate Geometry Revision Guide

Calculus and coordinate geometry are two of the most heavily examined areas of A-Level Pure Mathematics. Together they typically account for a significant proportion of Papers 1 and 2, and questions on these topics appear in some form almost every year. If you want a strong grade, you need to be confident in both.

These topics build on GCSE ideas -- gradients, equations of lines, areas under curves -- but extend them considerably. At A-Level you are expected to differentiate and integrate a wide range of functions, work with parametric and implicit equations, and solve problems that combine coordinate geometry with calculus. This guide covers everything you need, organised by topic, with advice on common mistakes and exam technique.

Differentiation

Differentiation is the backbone of A-Level calculus. You need to be fluent with a range of techniques and confident in applying them to different types of problem.

Differentiation from First Principles

The specification requires you to understand the limit definition of the derivative. You should be able to show that the derivative of x^n can be found by evaluating the limit of (f(x + h) - f(x)) / h as h tends to zero. Exam questions on first principles are usually restricted to simple functions -- x^2, x^3, or sin(x) -- but you must understand the underlying concept.

Standard Derivatives

You must memorise the derivatives of the core functions:

• y = x^n gives dy/dx = nx^(n-1) (the power rule).
• y = e^(kx) gives dy/dx = ke^(kx).
• y = a^(kx) gives dy/dx = ka^(kx) ln(a).
• y = ln(x) gives dy/dx = 1/x.
• y = sin(x) gives dy/dx = cos(x).
• y = cos(x) gives dy/dx = -sin(x).
• y = tan(x) gives dy/dx = sec^2(x).

The AQA formula booklet provides derivatives for sec(x), cosec(x), cot(x), and the inverse trig functions, but the results above must be in your memory. For a full breakdown of what is and is not in the booklet, see our formula booklet guide.

The Chain Rule, Product Rule, and Quotient Rule

These three rules are the tools you use to differentiate more complex expressions.

Chain rule: If y = f(g(x)), then dy/dx = f'(g(x)) x g'(x). You differentiate the outer function and multiply by the derivative of the inner function. The chain rule is used whenever you differentiate a composite function such as sin(3x^2), e^(x^2 + 1), or ln(2x + 5).

Product rule: If y = uv, then dy/dx = u(dv/dx) + v(du/dx). Use this when two functions are multiplied together, such as x^2 sin(x) or e^x ln(x).

Quotient rule: If y = u/v, then dy/dx = (v(du/dx) - u(dv/dx)) / v^2. Use this when one function is divided by another. Note that many quotient rule problems can also be solved using the product rule with a negative index.

Implicit Differentiation

Not all curves are given as y = f(x). When a relationship between x and y is given implicitly -- for example x^2 + y^2 = 25 -- you differentiate each term with respect to x, applying the chain rule to any term involving y. Each time you differentiate y with respect to x, you introduce a factor of dy/dx. You then rearrange to make dy/dx the subject.

Stationary Points and Optimisation

A stationary point occurs where dy/dx = 0. To classify it, use the second derivative test: if d^2y/dx^2 > 0 the point is a minimum, if d^2y/dx^2 < 0 it is a maximum. If the second derivative is zero, examine the sign of dy/dx on either side -- the point may be an inflection.

Optimisation problems require you to form an expression for the quantity to be maximised or minimised, differentiate, set the derivative equal to zero, and solve. These questions often involve setting up the problem from a worded context, which is where many marks are lost.

Related Rates of Change

These problems connect the rate of change of one variable to another using the chain rule: dy/dt = (dy/dx) x (dx/dt). For example, if you know the rate at which the radius of a sphere is increasing and you want the rate of volume increase, you differentiate V with respect to r and multiply by dr/dt.

Integration

Integration is the reverse of differentiation, but at A-Level it becomes a powerful topic in its own right, with several techniques you must master.

Standard Integrals

You need to memorise the reverse of the standard derivatives:

• Integral of x^n dx = x^(n+1)/(n+1) + c, for n not equal to -1.
• Integral of 1/x dx = ln|x| + c.
• Integral of e^(kx) dx = (1/k)e^(kx) + c.
• Integral of sin(kx) dx = -(1/k)cos(kx) + c.
• Integral of cos(kx) dx = (1/k)sin(kx) + c.

The formula booklet provides integrals of sec^2(kx), tan(x), cot(x), cosec(x), and sec(x). You should be familiar with these but will usually look them up during the exam.

Integration by Substitution

Substitution is the integration equivalent of the chain rule. You replace a complicated part of the integrand with a new variable u, express dx in terms of du, and integrate the simpler expression. In many exam questions the substitution is given, but you should also be able to spot the right substitution yourself. Common patterns include substituting the inner function of a composite expression, or using a trigonometric substitution for integrands involving square roots.

Integration by Parts

Integration by parts is derived from the product rule. The formula is: integral of u(dv/dx) dx = uv - integral of v(du/dx) dx. You need to choose u and dv/dx wisely. A useful mnemonic is LIATE (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) -- the function that comes first in this list is usually the best choice for u. Some integrals require you to apply the method twice, and some (such as the integral of e^x sin(x)) require applying it twice and then solving for the original integral.

Partial Fractions and Integration

When the integrand is a rational expression, you can often split it into partial fractions and integrate each fraction separately. For example, (3x + 5)/((x + 1)(x + 2)) can be written as A/(x + 1) + B/(x + 2), and each term integrates to a logarithm. You may also encounter repeated factors and improper fractions, in which case polynomial long division is needed first.

Definite Integrals and Areas

A definite integral gives the signed area between the curve and the x-axis between two limits. If the curve is below the x-axis the integral gives a negative value, so when calculating areas you need to split the integral at the roots and take the absolute value of each part. The area between two curves is found by integrating the difference of the two functions between the points of intersection.

The Trapezium Rule

The trapezium rule approximates a definite integral numerically. The formula is given in the AQA formula booklet. You need to know how to apply it and how to determine whether the approximation is an overestimate or underestimate by considering the concavity of the curve.

Volumes of Revolution

When a curve is rotated 360 degrees around the x-axis, the volume is V = pi x integral of y^2 dx. Around the y-axis, V = pi x integral of x^2 dy. You need to rearrange the equation of the curve to express y^2 or x^2 in terms of the appropriate variable before integrating.

Differential Equations

At A-Level the main technique is separation of variables. If you have dy/dx = f(x)g(y), you rearrange to get (1/g(y)) dy = f(x) dx and integrate both sides. The general solution contains an arbitrary constant c. If initial conditions are given (for example, y = 3 when x = 0), you find the particular solution by substituting the known values.

Differential equations frequently appear in modelling contexts -- exponential growth and decay (dP/dt = kP), Newton's law of cooling (dT/dt = -k(T - T_s)), and filling or emptying tanks. The key is to set up the equation correctly from the context, solve it, and interpret the solution.

Coordinate Geometry of Straight Lines

This topic builds directly on GCSE material but is tested in more demanding contexts.

You should be fluent with the following:

• Gradient: m = (y2 - y1) / (x2 - x1).
• Midpoint: ((x1 + x2)/2, (y1 + y2)/2).
• Distance between two points: d = sqrt((x2 - x1)^2 + (y2 - y1)^2).
• Equation of a line: y - y1 = m(x - x1), or y = mx + c.
• Parallel lines: Same gradient.
• Perpendicular lines: Product of gradients = -1.

At A-Level, coordinate geometry is often combined with other topics. You might need to find the equation of a tangent or normal to a curve using differentiation, or find where a line intersects a circle by substitution and solving simultaneously.

Circle Geometry

The equation of a circle with centre (a, b) and radius r is (x - a)^2 + (y - b)^2 = r^2. The expanded form is x^2 + y^2 + Dx + Ey + F = 0, and you need to be able to complete the square to convert back to centre-radius form.

Key results and techniques:

• Tangent to a circle: The tangent at a point is perpendicular to the radius at that point. Find the gradient of the radius, then use the negative reciprocal for the tangent gradient.
• Normal to a circle: The normal at a point on a circle passes through the centre.
• Condition for a line to be tangent to a circle: Substitute the equation of the line into the equation of the circle. If the resulting quadratic has a discriminant of zero, the line is a tangent.
• Perpendicular from centre to chord: The perpendicular from the centre to a chord bisects the chord. This is useful for finding chord lengths or distances from the centre.
• Intersection of a line and a circle: Substitute the line equation into the circle equation and solve the resulting quadratic. Two solutions means two intersection points, one means tangency, and no real solutions means no intersection.

Parametric Equations

Some curves are described using a parameter, usually called t or theta. Instead of a single equation relating x and y, you have two equations: x = f(t) and y = g(t).

Why Use Parametric Equations

Parametric equations are useful for curves that cannot easily be written as y = f(x) -- for example, a circle can be written as x = r cos(t), y = r sin(t). They also appear naturally in mechanics (where t represents time) and in the description of curves like the cycloid.

Converting Between Parametric and Cartesian Forms

To convert from parametric to Cartesian form, eliminate the parameter. This often involves rearranging one equation to make t the subject and substituting into the other, or using a trigonometric identity (for example, if x = cos(t) and y = sin(t), then x^2 + y^2 = 1).

Finding Gradients of Parametric Curves

The gradient of a parametric curve is given by dy/dx = (dy/dt) / (dx/dt). Differentiate x and y separately with respect to t and divide. This is one of the most commonly examined parametric techniques.

Areas Under Parametric Curves

The area under a parametric curve is given by the integral of y(dx/dt) dt. Replace the x-limits with the corresponding t-limits and integrate the product of y and dx/dt with respect to t.

Integration with Parametric Equations

Beyond finding areas, you may be asked to find the volume of revolution of a parametric curve or evaluate other integrals. The key principle is to express everything in terms of the parameter and the appropriate differential before integrating.

Common Mistakes

Here are the most frequent mistakes students make in these topics:

Forgetting the chain rule. When differentiating composite functions like sin(3x) or e^(x^2), students often forget to multiply by the derivative of the inner function. This is the single most common error in differentiation.

Incorrect signs in integration. The integral of sin(x) is -cos(x) + c, not cos(x) + c. Sign errors with trigonometric integrals are extremely common and cost multiple marks because the error carries through.

Not choosing the right technique. When faced with an integral, you need to decide between substitution, integration by parts, partial fractions, or a standard result. Practising a wide variety of integrals is the only way to build this judgement.

Errors converting parametric to Cartesian. Algebraic errors when eliminating the parameter are common, especially when trigonometric identities are involved. Write out every step carefully.

Forgetting +c for indefinite integrals. This is a freely available mark that students throw away every year. If there are no limits, include +c.

Miscalculating areas. When a curve dips below the x-axis the integral is negative. If you are asked for an area, split the integral at the roots and take absolute values.

Mixing up tangent and normal. The tangent has the same gradient as the curve at a point. The normal is perpendicular to it. Confusing the two wastes time and loses marks.

Exam Technique

AQA calculus and coordinate geometry questions tend to follow a structured format. Here is how to approach them:

Use earlier parts to help with later parts. Questions often build through parts (a), (b), (c). A result you prove in part (a) is almost certainly needed later. If you cannot complete an earlier part, state the result and use it anyway -- you will still earn marks for the later parts.

Show clear working for every step. Calculus questions require you to show the method, not just the answer. Write out each differentiation or integration step. If you use substitution, state it clearly. If you use integration by parts, show u, dv/dx, v, and du/dx.

State what you are differentiating with respect to. In parametric questions, make it clear whether you are finding dx/dt, dy/dt, or dy/dx. Ambiguity costs marks.

Check your answers. If you differentiate, check by integrating back. If you find a tangent equation, verify the point lies on both the tangent and the curve. Quick checks catch errors before you move on.

Manage your time. The longer calculus questions carry more marks but take longer. Make sure you have answered the shorter questions efficiently so that you have time for the extended problems.

Prepare with LearningBro

LearningBro offers several courses designed to build your confidence in these topics. The Calculus Applications course covers differentiation, integration, and differential equations with structured practice. The Coordinate Geometry in Depth course works through circles, parametric equations, and their intersection with calculus.

For broader coverage, the Pure Maths 1 and Pure Maths 2 courses follow the AQA specification topic by topic, and the AQA Exam Preparation course focuses on past paper practice and exam technique. Each course uses spaced repetition to help you retain key methods, so you walk into the exam with the techniques locked in.

Good luck with your revision.