Differentiation and integration — calculus — is the heaviest and most decisive topic in the first year of A level Maths. In the AS Pure paper it takes more than 30 hours of teaching and a large share of the marks. A pupil who understands what a derivative is, can find stationary points, can integrate a simple polynomial and can tackle an optimisation problem is set for a high grade. A pupil who cannot will struggle even if they are strong in geometry. This guide explains every tool you need from the first step: what a derivative is, the rules of differentiation for Year 12, how to analyse a curve, how to find an area with an integral, and how to attack an optimisation problem without panicking.
Differentiation and integration — calculus — is the topic that decides who gets a C and who gets an A in A level Maths. In the first year of the course (AS, Year 12) more than 30 hours are given to the calculus of polynomials: about 20 hours to differentiation and curve analysis, and another ten or so to integration. On the AS Pure paper, usually two whole questions are given to curve analysis and optimisation, which means a substantial share of the marks. At GCSE Higher your child met a taste of this — estimating the gradient of a curve by drawing a tangent, and the area under a speed–time graph — but Year 12 is where the ideas become exact. A pupil who puts the topic off and waits for Year 13 finds themselves starting the A2 year with a huge backlog, because everything that follows (the chain, product and quotient rules, exponentials, trigonometric functions) assumes you already live and breathe derivatives of polynomials. This guide is the complete road map: what a derivative is and why it is conceptually hard, which rules of differentiation you must know by heart, how to analyse a curve step by step, what an integral is and how it is calculated, how to attack optimisation problems — the weak point of most pupils — and finally, a six-week practice plan that takes you from zero to full command.
What is a derivative and why is it the hardest idea?
A derivative is the rate at which a function changes. If f(x) describes a person's height at age x, then f'(x) — the derivative — is the growth rate: how many centimetres they grow per year. If s(t) describes a car's position at time t, then s'(t) is its speed. The idea is completely ordinary from everyday life, but the formal definition — the limit of the difference quotient, lim[h→0] (f(x + h) − f(x))/h, called differentiation from first principles — is where most pupils get confused for the first time.
Why is it hard? Because three new ideas arrive at once: (1) a function as input–output (this was already at GCSE), (2) the idea of a limit — which is not a number but a process, and (3) the derivative itself, which is a new function derived from the first. In Year 12 the teacher covers first principles briefly, then gives the algebraic rules and asks you to practise them. That is the right approach — first get a feel for the calculation, then understand it in depth.
The geometric meaning: the derivative f'(x₀) is the gradient of the tangent to the graph of the function at the point (x₀, f(x₀)). If f'(3) = 2, that means at the point where x = 3 the graph is rising at a rate of 2 vertical units for every horizontal unit. That is why differentiation and linear functions are closely linked: the gradient of the straight line y = mx + c is m, and indeed the derivative of mx + c is m.
Notation: there are three accepted notations for the derivative — f'(x), dy/dx and y'. A level papers use both f'(x) and dy/dx. When you see d/dx[something], it simply means 'differentiate the something with respect to x'.
The rules of differentiation for Year 12
In the AS year the calculus is essentially about polynomials and other powers of x — there is no differentiating sin, cos, ln or eˣ yet (those arrive in Year 13, along with the chain, product and quotient rules). That means you only need four rules by heart, and that is it.
Rule 1 — the power rule: the derivative of xⁿ is n·xⁿ⁻¹. Examples: the derivative of x² is 2x. The derivative of x³ is 3x². The derivative of x⁷ is 7x⁶. The derivative of x itself (that is x¹) is 1·x⁰ = 1. The rule also works for negative and fractional powers, which AS does use: the derivative of x⁻² is −2x⁻³, and the derivative of x^(1/2) is ½x^(−1/2).
Rule 2 — the derivative of a constant is zero. The derivative of 5 is 0. The derivative of −π is 0. The derivative of √2 is 0. The reason: a constant does not change, so its rate of change is always zero.
Rule 3 — a constant times a function. The derivative of c·f(x) is c·f'(x). The constant simply stays where it is. The derivative of 7x² is 7·(2x) = 14x. The derivative of −3x⁴ is −3·(4x³) = −12x³.
Rule 4 — the derivative of a sum or difference. The derivative of f(x) ± g(x) is f'(x) ± g'(x). Simply differentiate each term separately. The derivative of x³ + 5x² − 4x + 7 is 3x² + 10x − 4 + 0 = 3x² + 10x − 4.
A combined example: differentiate f(x) = 2x⁴ − 3x³ + 6x² − 9x + 11. Term by term: f'(x) = 2·(4x³) − 3·(3x²) + 6·(2x) − 9·1 + 0 = 8x³ − 9x² + 12x − 9. Notice: every term dropped one power, and every constant vanished.
An important trick — the second derivative. The second derivative f''(x) (or d²y/dx²) is simply the derivative of the derivative. In the previous example, f'(x) = 8x³ − 9x² + 12x − 9, so f''(x) = 24x² − 18x + 12. The second derivative is used in curve analysis to classify maxima and minima and to find points of inflection.
Analysing a curve — what is it and what do you check?
Analysing a curve means fully identifying the behaviour of f(x): where it is defined, where it crosses the axes, where it is increasing and where decreasing, and what its stationary points are. A typical exam question gives you a polynomial and asks you to find its stationary points, determine their nature and sketch the graph. The steps are always the same — learn them once, and they always work.
Step 1 — the domain. For a polynomial the domain is always all real numbers (there is no root to check, no denominator to check). Simply write 'x ∈ ℝ'. But do not skip this step — for functions with a denominator or a root later in the course, it matters, and getting into the habit now costs nothing.
Step 2 — intercepts with the axes. With the y-axis: substitute x = 0 and find f(0). With the x-axis: solve the equation f(x) = 0. For a quadratic use factorising or the quadratic formula. For a cubic or quartic, try taking out a common factor or spotting a root by inspection (x = 1, x = −1, x = 2 are good candidates), then use the factor theorem.
Step 3 — increasing and decreasing. Differentiate the function and solve f'(x) = 0. That gives the candidates for stationary points. Build a sign table for f'(x) — mark the roots of the derivative on an x-axis, and in each interval check whether the derivative is positive (so f is increasing) or negative (so f is decreasing).
Step 4 — stationary points. A point where f'(x) changes sign from + to − is a local maximum. A point where the sign changes from − to + is a local minimum. A point where the derivative is zero but does not change sign (for example x³ at x = 0) is a stationary point of inflection, not a maximum or minimum.
A full example: f(x) = x³ − 3x² + 4. Domain: all of ℝ. y-intercept: f(0) = 4, so (0, 4). x-intercepts: x³ − 3x² + 4 = 0; spot a root — x = −1 works because −1 − 3 + 4 = 0. Differentiate: f'(x) = 3x² − 6x = 3x(x − 2). Roots of the derivative: x = 0 and x = 2. Sign table: on (−∞, 0) the derivative is positive (f increasing), on (0, 2) the derivative is negative (f decreasing), on (2, ∞) the derivative is positive again (f increasing). So x = 0 is a local maximum with value f(0) = 4, and x = 2 is a local minimum with value f(2) = 8 − 12 + 4 = 0.
Step 5 — sketching the graph. Mark the intercepts with the axes and the stationary points, and draw the curve according to where it is increasing and decreasing. The sketch does not have to be accurate to scale — it has to be qualitatively correct.
Integration and area
If the derivative is 'the rate of change', the integral is the reverse operation — given the rate, get back to the original function. And in its geometric meaning: a definite integral calculates the area under the graph of the function. In the AS year you integrate polynomials and other powers of x, about 10 hours of teaching.
The indefinite integral — the antiderivative. If f'(x) = 2x, then f(x) = x² + c, where c is some constant. Notation: ∫2x dx = x² + c. The c is critical — without it you lose a mark in the exam. The reason: every function of the form x² + 7, x² − 3, x² + 100 has the same derivative 2x.
The power rule for integration: ∫xⁿ dx = xⁿ⁺¹/(n + 1) + c. Examples: ∫x³ dx = x⁴/4 + c. ∫x⁵ dx = x⁶/6 + c. The rule does not work for n = −1 (because then there is a division by zero) — the integral of 1/x is ln x, which arrives in Year 13.
Integrating a sum and a constant multiple: as with differentiation, work term by term. ∫(3x² − 4x + 5) dx = 3·(x³/3) − 4·(x²/2) + 5x + c = x³ − 2x² + 5x + c.
The definite integral — calculating an area. ∫[a to b] f(x) dx = F(b) − F(a), where F is any antiderivative of f. Notice: in a definite integral the constant c cancels (because it appears twice and subtracts out), so there is no need to add c when evaluating a definite integral.
Example: evaluate ∫[1 to 3] x² dx. The antiderivative is x³/3. Substitute: (3³/3) − (1³/3) = 27/3 − 1/3 = 26/3. The area under the parabola y = x² between x = 1 and x = 3 is 26/3 square units.
Care with sign: a definite integral gives a 'signed' area. If the graph is below the x-axis, the integral comes out negative, and the actual area is the absolute value. When a question asks for 'the area' — always take the absolute value, or split the integral into sections and flip the sign on any section where the function is negative.
Area between two curves: ∫[a to b] (f(x) − g(x)) dx, where f is the 'upper' function and g the 'lower' one over the interval. The roots of the equation f(x) = g(x) are the points of intersection and set the limits of integration.
Optimisation problems — where most pupils fall down
An optimisation problem is a worded question asking for the maximum or minimum value of a geometric or physical quantity. Examples: 'Of all rectangles with perimeter 40 cm, which has the largest area?', or 'What height of a one-litre tin gives the smallest surface area?'. In the exam this question loses a lot of marks because it needs a combination of algebra, geometry, differentiation and interpreting a worded problem — and many pupils do not know where to start.
The fixed recipe — 5 steps. (1) Identify the variable — what changes in the problem? Call it x. (2) Express the quantity to be maximised or minimised as a function of x — often you need a constraint equation to eliminate a second variable. (3) State the allowed range of x (for example, if x is a length, then x > 0). (4) Differentiate and solve f'(x) = 0. (5) Check that it really is a maximum or minimum (sign table or second derivative), and give the values of every variable in the problem.
Example — a rectangle with perimeter 40. Recall: of all rectangles with perimeter 40 cm, which has the largest area? Let the shorter side be x. The perimeter constraint: 2(x + y) = 40, so y = 20 − x. Range: 0 < x < 20. The area: A(x) = x·y = x(20 − x) = 20x − x². Differentiate: A'(x) = 20 − 2x. Solve A'(x) = 0: x = 10. Second derivative: A''(x) = −2 < 0 → it is indeed a maximum. So x = 10 and y = 10, in other words a square, and the area is 100 cm². Conclusion: of all rectangles with a given perimeter, the square has the largest area.
Common mistakes in optimisation: (a) forgetting to write the constraint equation and ending up with two variables in the derivative — everything must be expressed in one variable. (b) Forgetting to check that it is a maximum rather than a minimum (this is also a source of logical errors in the question). (c) Forgetting to return the answer to the language of the problem — if the question asked for the dimensions, you must give both x and y, not only x.
A six-week practice plan
This plan assumes about an hour a day, 5 days a week. It is built to start from someone who knows the topic only superficially and take them to exam level. If you are already in Year 13 and preparing for an exam in three months, double the pace — two weeks per stage.
Week 1 — technical differentiation. Only the rules of differentiation. 30 differentiation questions on polynomials, until each one is differentiated in one go without thinking. Do not move on until you can differentiate 5x⁴ − 3x² + 7 in 10 seconds in your head.
Week 2 — analysing quadratic and cubic curves. Practise the five steps of curve analysis on parabolas and on cubics. One complete function a day, with a hand sketch of the graph.
Week 3 — quartics and beyond. By now the differentiation is automatic, but solving f'(x) = 0 when it is a cubic sometimes needs a common factor or an integer root by inspection. At this stage also add the second derivative and points of inflection.
Week 4 — indefinite and definite integrals. Learn the power rule for integration, and work through 20 questions on the area under a curve. This chapter is much more technical and shorter — if your differentiation is secure, the integration flows.
Week 5 — optimisation problems. This is where to invest. Practise at least 10 worded problems of every kind: geometric (rectangle, cylinder, cone), financial (profit, cost), and distance/speed. Work through the 5 steps deliberately each time.
Week 6 — full exam questions. Take at least 5 complete questions from past AS papers (any of the boards — Edexcel, AQA, OCR or OCR MEI — the calculus content is the same), answer them under exam conditions (timed, without looking at the solution) and mark afterwards. At this stage the weaknesses that remain should be only technical — minus signs, forgetting c — not conceptual.
Summary, and where next
Differentiation and integration is the one topic in A level Maths that runs from the first weeks of Year 12 to the last paper of Year 13. Serious investment now, while everything is still well-behaved polynomials, is the most rewarding investment you can make in sixth-form maths. A pupil who in the 30 hours of Year 12 builds a reflex for differentiating, analysing a curve and solving optimisation problems — everything that follows (the chain rule, logarithms, sines, exponentials) is just swapping in one more rule. A pupil who does not — has to learn both the idea and the technique at the same time in Year 13, which is double the load.
Frequently asked questions
How many hours of teaching go to calculus in AS Maths?
In a typical AS scheme of work: about 20 hours to differentiation of polynomials and curve analysis, and another 10 or so to integration — 30+ hours out of the Pure content in Year 12. That makes calculus the heaviest topic of the year, more than trigonometry and more than statistics.
Do you need to know the derivative of sin, cos or ln in Year 12?
No. The AS year limits calculus to powers of x — polynomials, negative and fractional indices. Derivatives of trigonometric, exponential and logarithmic functions, along with the chain, product and quotient rules, are Year 13 (A2) content. Concentrating on polynomials in the first year lets the idea settle fully before the complexity is added.
What is the difference between a derivative and an integral?
A derivative is the rate of change of a function — on the graph, it is the gradient of the tangent at each point. An integral is the reverse operation: given the rate, it reconstructs the original function, and in its geometric meaning it calculates the area under the graph. Differentiation and integration are a pair of inverse operations, exactly like addition and subtraction, or powers and roots.
Why must you add + c to an indefinite integral?
Because every function has infinitely many antiderivatives, differing only by a constant. The derivative of x² + 5 and the derivative of x² − 100 are both 2x. The c in x² + c represents the whole family. In the exam you lose a mark for forgetting the c in an indefinite integral. In a definite integral the c cancels automatically, so there you do not have to write it.
How do you know whether a stationary point is a maximum or a minimum?
Two methods. Method 1 (sign table): check the sign of f'(x) on each side of the point. If it changes from + to −, it is a maximum. From − to +, a minimum. If it does not change, it is a stationary point of inflection. Method 2 (second derivative): substitute the x of the stationary point into f''. If f'' < 0 — maximum. If f'' > 0 — minimum. If f'' = 0 — the test is inconclusive and you go back to the sign table.
What share of the AS Pure paper is calculus?
On the AS Pure paper, usually about a quarter to a third of the marks involve differentiation or integration — typically one curve-analysis or tangent question and one optimisation or area question, depending on the paper. On the A2 papers calculus grows further, because it combines with every other topic. A pupil who is not secure with derivatives will find the second year very hard to pass.
Plot a polynomial, read its gradient and watch the stationary points appear
Explore a curve and its tangent ←