Ten Oxford and Cambridge questions built from Newton's laws to momentum conservation, differentiation from first principles, the double-slit equation, a spring's simple harmonic motion, and Snell's law from Huygens' principle — each with a full step-by-step model answer written by specialist subject tutors.
Momentum conservation rebuilt from Newton's laws, a derivative found from first principles, Snell's law reached from Huygens' principle instead of recited: the Derivations pack treats results you already know as things to prove again from scratch. The double-slit and SHM equations get the same treatment, and so does the full SUVAT set — ten arguments taken all the way from a first statement to a finished proof, never handed over as a formula sheet.
The Derivations pack — £180
Ten questions across 30 pages, each one building in three stages: the question alone, then the hints if you need them, then the full worked answer. One PDF, one payment, instant download.
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Derivation questions are among the most intellectually revealing question types in Oxford and Cambridge Maths, Further Mathematics, and Physics interviews. They ask you not just to reproduce a result but to construct the logical argument that establishes it — to reason step by step from a starting point to a conclusion, explaining what each step achieves and why it is valid. This skill — the ability to derive rather than merely recall — is fundamental to the undergraduate experience at both universities, and demonstrating it in the interview is one of the strongest signals a candidate can send. Of the five Maths packs, Derivations is the one built entirely around that skill: ten questions, split evenly between pure maths and physics, that move from Newton's laws to momentum conservation, from a first-principles derivative to a self-referential integral, and from Huygens' principle to Snell's law, each one asking for the argument before it asks for the answer.
When an Oxford or Cambridge interviewer asks you to derive a result, they are testing something that is impossible to assess with a factual question: whether you understand why a result is true, rather than just knowing that it is. This distinction is fundamental to how mathematics and physics are taught and practised at undergraduate level. At A-level, you are often given formulae to apply; at Oxbridge, you are expected to understand where those formulae come from and to be able to reconstruct them from first principles when needed.
Derivation questions also reveal mathematical maturity in a way that calculation questions cannot. A student who can derive the formula for the sum of a geometric series — starting from the definition, multiplying through by the common ratio, subtracting, and obtaining a closed form — has demonstrated that they understand the algebraic structure underlying the formula, not just its numerical consequences. A student who can derive the lens equation from the geometry of refraction rather than just applying 1/f = 1/u + 1/v has demonstrated that they understand the physics, not just the calculation procedure.
Interviewers use derivation questions to reveal this depth of understanding in real time. They can observe whether you can identify the starting point of a derivation, construct each step with appropriate justification, and arrive at the correct conclusion — and they can introduce complications (what if the series is infinite? what if the lens is immersed in water?) to test whether your understanding extends beyond the specific case you derived.
The approach to derivation questions that works reliably in Oxford and Cambridge interviews has four components. The first is orientation: before writing anything down, state what you are trying to derive and what you are allowed to assume. This clarifies the starting point and demonstrates that you understand the structure of the argument before you begin. It also prevents the common error of starting in the middle of a derivation without establishing what is given.
The second component is stepwise construction with narration. Each step of the derivation should be accompanied by a verbal explanation of what you are doing and why: 'I'm going to multiply both sides by r, the common ratio, because that will give me an expression I can subtract from the original sum and most of the terms will cancel.' This narration is crucial because it demonstrates understanding rather than memorised execution — and it allows the interviewer to follow your reasoning and intervene helpfully if you go off track.
The third component is dimensional checking. At each stage of a physics derivation, check that your expression has the correct dimensions. If you are deriving the period of a pendulum and your intermediate expression has dimensions of velocity rather than time, you have made an error and can identify it before it propagates. Dimensional analysis is a powerful self-checking tool that interviewers regard highly.
The fourth component is the limiting case check. Once you have arrived at a result, check it against known limiting cases. The formula for the period of a simple pendulum should reduce to something physically sensible as the length approaches zero; the formula for the kinetic energy of a relativistic particle should reduce to (1/2)mv² when v is much less than c. Performing these checks unprompted demonstrates physical and mathematical intuition of a high order, and it is the most effective way to distinguish yourself in a derivation question.
The lists below are the derivations tested most consistently across Oxford and Cambridge interviews in general — not a description of this pack. The ten questions actually inside the Derivations pack are set out in the FAQ further down this page.
Several derivations appear sufficiently consistently in Oxford and Cambridge Maths interviews that every applicant should be able to produce them fluently and from first principles. The most important are as follows.
The sum of a geometric series: this derivation is elegant and short. Let S = a + ar + ar² + ... + arⁿ⁻¹. Multiply by r: rS = ar + ar² + ... + arⁿ. Subtract: S(1-r) = a(1-rⁿ). Divide: S = a(1-rⁿ)/(1-r). The key insight — multiplying by r to create a telescoping subtraction — should be explained explicitly, not just executed. For the infinite series where |r| < 1, take n → ∞ to get S = a/(1-r).
Integration by parts: this follows directly from the product rule. If u and v are functions of x, then d(uv)/dx = u·dv/dx + v·du/dx. Rearranging: u·dv/dx = d(uv)/dx − v·du/dx. Integrating both sides with respect to x: ∫u(dv/dx)dx = uv − ∫v(du/dx)dx. The key step — rearranging the product rule and integrating — should be stated explicitly, because the derivation makes clear why integration by parts is a valid technique rather than just a formula to memorise.
The quadratic formula: complete the square on ax² + bx + c = 0. Divide by a: x² + (b/a)x + c/a = 0. Complete the square: (x + b/2a)² − b²/4a² + c/a = 0. Rearrange: (x + b/2a)² = (b² − 4ac)/4a². Take the square root: x + b/2a = ±√(b² − 4ac)/2a. Solve: x = (−b ± √(b² − 4ac))/2a. Each algebraic step should be stated and justified — this is a derivation, not just a computation.
| Derivation | Subject | Key Insight |
|---|---|---|
| Sum of geometric series | Maths | Multiply by r, subtract to telescope |
| Integration by parts | Maths | Rearrange product rule and integrate |
| Quadratic formula | Maths | Complete the square on general ax²+bx+c=0 |
| Period of simple pendulum | Physics | Small-angle approximation: sin θ ≈ θ |
| Escape velocity | Physics | Energy conservation: KE = gravitational PE |
| Energy stored in capacitor | Physics | Integrate work done dW = V dq over charging |
| Lens equation 1/f = 1/u + 1/v | Physics | Similar triangles in refraction geometry |
| Kinematic equations (SUVAT) | Physics | Integrate acceleration; apply boundary conditions |
Physics derivation questions test a distinct set of skills from Maths derivations: not just algebraic manipulation but the ability to identify the correct physical starting point, set up the problem correctly using appropriate idealisation, and apply conservation laws or differential equations to reach a result. The following are the most important derivations for Oxford and Cambridge Physics interviews.
The period of a simple pendulum: the key insight is the small-angle approximation sin θ ≈ θ (valid for θ in radians when θ is small). The restoring torque on the pendulum bob is −mgL sin θ ≈ −mgLθ for small θ, giving the equation of motion d²θ/dt² = −(g/L)θ — which is simple harmonic motion with angular frequency ω = √(g/L) and period T = 2π√(L/g). Interviewers frequently follow this with: 'What happens to the period if the amplitude is not small?' — which requires knowing that the period increases monotonically with amplitude, approaching infinity as the amplitude approaches 180°.
Escape velocity: the minimum launch speed for which a projectile escapes a planet's gravitational field is found by energy conservation. Set the total mechanical energy at launch equal to zero (the minimum energy to just reach infinity): ½mv² − GMm/R = 0, where M is the planet's mass and R its radius. Solving: v = √(2GM/R). Equivalently, v = √(2gR) where g is the surface gravitational acceleration. Checking: this gives approximately 11.2 km/s for Earth, which matches the known value. The derivation should state explicitly that the escape velocity is independent of the mass of the projectile, which is a consequence of the equivalence principle.
Energy stored in a capacitor: the work done to add an infinitesimal charge dq to a capacitor at potential V = q/C is dW = V dq = (q/C) dq. Integrating from 0 to Q: W = ∫₀^Q (q/C) dq = Q²/(2C) = ½CV². The key step — setting up the integral of work against a varying potential — should be explained explicitly. Interviewers sometimes follow this with a variation: 'Two charged capacitors are connected in parallel — how much energy is lost?' — which needs charge conservation as well as energy accounting.
More demanding derivation questions in Oxbridge interviews introduce complications that require you to extend a standard derivation beyond its familiar form. These questions test not just whether you have memorised a derivation but whether you understand it well enough to modify it when the situation changes.
A common format is the modified physical system: instead of a simple pendulum, derive the period of a compound (physical) pendulum rotating about a pivot not at its centre of mass. This requires you to introduce the moment of inertia I about the pivot, write the equation of motion as Iα = −mghθ (where h is the distance from the pivot to the centre of mass), and identify ω = √(mgh/I) — the same functional form as the simple pendulum but with different physical quantities. A student who understands the derivation of the simple pendulum period can construct this in a few minutes; a student who has only memorised the result cannot.
Another common format is the dimensionally constrained derivation: 'Without knowing the formula, derive the period of a simple pendulum using only dimensional analysis.' This requires identifying the relevant physical quantities (length L, gravitational acceleration g, mass m), noting that the period must have dimensions of time, and constructing the only dimensionally consistent combination: T ∝ √(L/g). The fact that mass drops out follows from dimensional analysis alone — which is a striking and physically significant result that interviewers use to introduce discussions of the equivalence principle.
Prove that logb(x) = 1 / logx(b) for any valid base b and any x > 0 with x ≠ 1. Then use this to help solve the equation (log3 x)² − 5(log3 x) + 6 = 0.
Let y = logb(x), so that by = x. Taking logx of both sides gives logx(by) = logx(x) = 1, and by the power rule for logarithms, logx(by) = y · logx(b). So y · logx(b) = 1, giving y = 1/logx(b) — and since y = logb(x), this proves logb(x) = 1/logx(b). For the equation, substituting u = log3 x turns it into u² − 5u + 6 = 0, which factorises as (u − 2)(u − 3) = 0, giving u = 2 or u = 3. Undoing the substitution: log3 x = 2 gives x = 9, and log3 x = 3 gives x = 27 — both valid, since x must be positive. Two of the Derivations pack's own ten questions follow this same shape: a logarithm rule proved from the definition, then an equation that looks unfamiliar until the right substitution turns it into a quadratic you already know how to solve.
For Further Mathematics and Mathematics applicants, derivation questions extend to formal proofs — particularly proof by mathematical induction. The structure of an induction proof is consistent across all problems: prove the base case (typically n = 1 or n = 0), assume the statement is true for n = k (the inductive hypothesis), and then show it must be true for n = k + 1. The conclusion is that the statement holds for all positive integers n.
The most common induction problems in Oxford and Cambridge interviews involve sums of series: prove that 1 + 2 + 3 + ... + n = n(n+1)/2 by induction; prove that the sum of the first n cubes equals [n(n+1)/2]²; prove that 2ⁿ > n² for all sufficiently large n. In each case, the base case is straightforward and the inductive step requires careful algebraic manipulation of the assumption to establish the conclusion.
A common mistake is to state 'assume true for n = k, therefore true for n = k + 1' without actually doing the algebraic work — which is tautological rather than a proof. The algebraic step is the substance of the induction: you must start from the assumed formula for k terms, add the (k+1)th term, and show algebraically that the resulting expression matches the formula with k+1 substituted in. Each line of algebra should be justified, and the conclusion should state explicitly that the inductive step has been completed. Induction proofs like these do not appear in the Derivations pack, which stays with first-principles derivations rather than formal proof — see the FAQ further down this page for exactly what its ten questions cover.
The three layers below are the ones this pack itself uses for every question: the bare question, then hints, then a full worked answer. What follows is not one of the pack's own ten questions — it uses a different function so nothing here gives away paid content — but it is worked in exactly the same order and to the same length, on the technique the pack's own differentiation question tests.
Using the definition of a derivative as a limit, derive d/dx(cos x) from first principles. You may assume the compound-angle formula cos(A + B) = cos A cos B − sin A sin B, and the two standard limits limh→0 sin(h)/h = 1 and limh→0 [cos(h) − 1]/h = 0.
By definition, d/dx(cos x) = limh→0 [cos(x + h) − cos(x)] / h. Expanding cos(x + h) = cos(x)cos(h) − sin(x)sin(h) and substituting: [cos(x)cos(h) − sin(x)sin(h) − cos(x)] / h = cos(x)·[(cos(h) − 1)/h] − sin(x)·[sin(h)/h]. Taking the limit of each bracket separately — using the two standard limits given in the question — the first bracket tends to 0 and the second tends to 1. So d/dx(cos x) = cos(x)·0 − sin(x)·1 = −sin(x).
Not the algebra, which is short once the compound-angle formula is expanded. It is testing whether a candidate can apply the limit definition of a derivative to a function they have not seen worked from first principles before, rather than simply quoting the result. One of the Derivations pack's own ten questions asks for exactly this — differentiation from first principles — and applies it twice, to sin x and then to x^x, with the worked answer offering two separate routes through the one limit that usually stalls candidates.
Both demonstrations above use techniques the Derivations pack tests directly — a logarithm identity folded into a quadratic, and a derivative found from first principles — worked here on examples built independently of the pack rather than lifted from it. The pack's own ten questions, evenly split between pure maths and physics, move from two logarithm-and-quadratic pairs and a pair of trigonometric identities through differentiation from first principles and integration by parts, to Newton's laws and momentum conservation, the double-slit equation, a spring's simple harmonic motion, the SUVAT equations, and Snell's law from Huygens' principle — each of the ten given first as a bare question, then a page of hints, then the full worked answer that follows it.
Two things to be straight about first. There is no Derivations sample to download, so this page is as far as the free material goes and nothing further is waiting behind a form. And the pack cannot mark you: a derivation quietly reverse-engineered from a formula you already knew looks, on the page, exactly like one built forwards. The free Maths sample is the nearest thing to a preview — its questions are Maths ones rather than derivations, but the model answers are written to the same standard.
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Get the Derivations pack — £180"I had no idea what to expect from my Maths interview at Magdalen — A-level gives you no preparation for the style of question they ask. Working through the pack beforehand meant I'd practised thinking through problems I'd never seen before and talking through my reasoning out loud. When I got stuck in the actual interview, I knew how to keep going rather than freeze."— James H., Mathematics, Magdalen College Oxford, 2024 entry
“My panel at Gonville & Caius handed me a short article about a clinical trial and asked what I thought the key limitation was. I’d never seen the paper before. The pack was the only preparation I found that actually trains you for that — reading through the model answers showed me how to reason about evidence out loud, identifying what is missing or uncertain rather than just summarising what is there. By the time I got into the room I knew how to think, not just what to say.”— Priya S., Medicine, Gonville & Caius Cambridge, 2024 entry
"My tutor pushed back on everything I said. The pack was the only resource I found that prepares you for that — the model answers show you how to structure an argument and defend it under pressure. Really glad I used it."— Ella T., History, Balliol College Oxford, 2025 entry
Ten questions, split evenly between pure maths and physics. Two turn a logarithm rule into a quadratic equation to solve — one built on ln x, one on e^-x. One derives differentiation from first principles and applies it twice, to sin x and separately to x^x. One states Newton's laws of motion, then builds a proof that momentum is conserved between two particles of any mass in a closed system. One derives the formula linking fringe spacing to wavelength for a double-slit setup, and applies it to a beam of light entering the slits at an angle. One derives the equation of motion for a mass on a spring, shows the motion is simple harmonic, and finds the period. One is a pair of trigonometric-identity questions. One derives integration by parts from the product rule and applies it to an integral that has to be solved for itself after a second pass. One derives all four SUVAT equations from a velocity-time graph, then sets a harder question where the acceleration is a function of displacement rather than time. The last derives Snell's law from Huygens' principle and applies it to an optical fibre's core and cladding.
By strand rather than by difficulty alone: two logarithm-into-quadratic pairs, one calculus derivation used on two different functions, one mechanics derivation building from Newton's laws to momentum conservation, one wave-optics derivation, one oscillations derivation, one trigonometric-identity pair, one calculus derivation built around a self-referential integral, one kinematics derivation that ends in a harder variable-acceleration question, and one further optics derivation moving from Huygens' principle to an application involving an optical fibre. The pack's own introduction says the difficulty rises through the ten questions and recommends starting near the front rather than dipping in partway through.
Past the headline result. The differentiation-from-first-principles question gives two separate routes to the one part of the limit that usually stalls candidates — an approximation-based argument and a proof using L'Hopital's rule — so a candidate who only knows one method still sees where the other would have led. The harder SUVAT question carries the algebra through two simultaneous equations to a numerical constant before taking the resulting expression to its limit at infinity, rather than stopping once a plausible-looking formula appears.
Both, in an even split. Five questions are pure maths — the two logarithm-and-quadratic questions, differentiation from first principles, the trigonometric-identity pair, and integration by parts — and five are physics: Newton's laws and momentum conservation, the double-slit equation, a spring's simple harmonic motion, the SUVAT equations, and Snell's law from Huygens' principle. It is built for Maths, Further Mathematics, or Physics applicants whose interviews are likely to move between the two within a single question.
No. Its ten questions are the ones above: logarithm identities and the quadratics they turn into, differentiation from first principles, Newton's laws and momentum, the double-slit equation, simple harmonic motion, trigonometric identities, integration by parts, the SUVAT equations, and Snell's law from Huygens' principle. There is no proof by induction and no series work anywhere in it.
A candidate who already knows the standard results — Newton's laws, the SUVAT equations, Snell's law — and needs practice constructing the argument that connects them rather than reciting the destination. The middle layer in this pack is Hints, not Prompts: short nudges written to read like the follow-up an interviewer gives when you are stuck on a specific step, rather than a running commentary meant to be read before you attempt the question.
Further Reading: Derivations sit at the heart of both Maths and Physics Oxbridge interviews. For broader context on the Oxford Maths interview format and worked examples, see our companion guide: Oxford Maths Interview Questions 2026 — Step-by-Step Model Answers.
See also: Maths Oxbridge Interview Questions, Graph Sketching Oxbridge Interview Questions, Integration & Curve Sketching Oxbridge Interview Questions, and MAT preparation.
Build each derivation with the model answer face down, out loud, writing the justification for a line before you write the line itself. Then compare — not just whether you reached the right formula, since several of these are results you already know before you start, but whether every step on the way there was one you could have justified out loud. Two of the techniques this pack tests — a logarithm identity folded into a quadratic, and a derivative found from first principles — are worked in full further up this page; the pack's own ten questions add Newton's laws and momentum conservation, the double-slit equation, a spring's simple harmonic motion, a pair of trigonometric identities, the SUVAT equations, and Snell's law from Huygens' principle. Derivations is one of the five Maths packs, and it is the one built entirely around constructing an argument rather than reciting one.
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