A Level · Cambridge (CIE) · 9709

Differentiate quotients and preserve the original denominator exclusions

Keep the subtraction order and squared denominator, then evaluate gradients or rates on the correct connected domain interval.

Section 2.4 · Differentiation

Pure Mathematics 2 & 3 · Differentiation · 2.4 / 3.4

Exam scope and assessment notes

Pure Mathematics 2 (Paper 2, AS only) and Pure Mathematics 3 (Paper 3, A Level) · Mathematics 9709, 2026–2027. Paper 1 knowledge is assumed. The AS route using Papers 1 + 2 cannot be carried forward to A Level. Shared sections 2.4 and 3.4 cover exponential, natural logarithmic and trigonometric derivatives, composites, products and quotients, and first derivatives of parametric and implicit curves, including tangents and normals. Paper 3 additionally requires the derivative of inverse tangent; use its separate lesson for that method. Derivatives of inverse sine and inverse cosine are not required in section 3.4. Trigonometric differentiation uses radians. Keep original domains and check zero derivative denominators separately. Written practice is self-checked.

Before you begin

Differentiate polynomial, log and radian trigonometric terms.
Find original denominator restrictions and use point-gradient line equations.

Work on paper. Try each question before revealing help, and explain why your method works.

Quick readiness check

What is the quotient-rule numerator for u/v?

What you will learn

Apply (u/v)′ = (u′v − uv′)/v² and simplify without changing the original domain.
Use quotient derivatives for curve lines or a supplied rate model and distinguish separate domain intervals.

Keep the numerator order and the denominator square

For y = u/v where v ≠ 0, the quotient rule is y′ = (u′v − uv′)/v². The numerator begins with the top derivative times the unchanged bottom, then subtracts the unchanged top times the bottom derivative. Reversing those two terms changes the sign. Deriving it as the product uv^(−1) also gives u′/v − uv′/v², the same rule on the original denominator domain.

For y = (2x − 4)/(3x + 2), the numerator is 2(3x + 2) − 3(2x − 4) = 16. Thus y′ = 16/(3x + 2)² for x ≠ −2/3. It is positive on each interval x < −2/3 and x > −2/3. This does not make the disconnected domain one unbroken increasing curve: crossing the excluded pole is not an interval on which the derivative argument applies.

The original expression determines allowed evaluation points

For y = sin x/x with radian x and x ≠ 0, the derivative is (x cos x − sin x)/x². A possible limiting extension at zero would be a separately defined function; the original quotient remains undefined there. For y = ln x/(x + 1), require x > 0, which also ensures the denominator is non-zero. Its derivative is [(x + 1)/x − ln x]/(x + 1)². At x = 1 the curve point is (1, 0) and its gradient is 1/2.

A supplied distance model q(t) = 10t/(t + 2) metres for t ≥ 0 seconds has derivative q′(t) = 20/(t + 2)² m/s. At t = 3, its distance is 6 metres and instantaneous speed is 4/5 m/s. The model's time domain is separate from the algebraic quotient exclusion t = −2; the non-negative domain already avoids that excluded input. Do not confuse a distance value with its speed.

Pause and explain

Since 16/(3x + 2)² is positive, what can be concluded about (2x − 4)/(3x + 2)?

Put the idea to work

Worked example

Differentiate y = (2x − 4)/(3x + 2) and state where the result applies.

Show the worked solution

The original denominator requires x ≠ −2/3; use u′ = 2 and v′ = 3.
Form the quotient numerator 2(3x + 2) − 3(2x − 4) = 16.
Divide by (3x + 2)² to obtain y′ = 16/(3x + 2)² on the original domain.

Answer y′ = 16/(3x + 2)², with x ≠ −2/3; increasing on each separate domain interval.

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From guided practice to a new situation

Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.

1 · Guided

Differentiate y = sin x/x in radians and state the original excluded input.

Give me a hint

Use u = sin x and v = x, retaining the quotient subtraction order.

Compare my reasoning

The original function requires x ≠ 0, and the derivatives are u′ = cos x and v′ = 1.
The quotient numerator is x cos x − sin x.
Hence y′ = (x cos x − sin x)/x² for x ≠ 0; the original quotient is not defined at zero.

y′ = (x cos x − sin x)/x², with x ≠ 0.

Look for these in your work

Apply the radian sine derivative and correct numerator order.
Retain the original exclusion after forming the derivative.

2 · Independent

Differentiate y = ln x/(x + 1) for x > 0 and find tangent and normal equations at x = 1.

Give me a hint

Use the quotient rule, then find the original point and evaluate the derivative there.

Compare my reasoning

The derivative is [(x + 1)/x − ln x]/(x + 1)² on x > 0.
At x = 1 the point is (1, 0) and the gradient is (2 − 0)/4 = 1/2.
The tangent is y = (x − 1)/2 and the normal is y = −2(x − 1).

y′ = [(x + 1)/x − ln x]/(x + 1)²; tangent y = (x − 1)/2; normal y = −2(x − 1).

Look for these in your work

Use the log derivative and original positive domain.
Build both lines from the original point with perpendicular gradients.

3 · Transfer

A supplied distance model is q(t) = 10t/(t + 2) metres for t ≥ 0 seconds. Find its distance and instantaneous speed at t = 3.

Give me a hint

Use the original quotient for distance and its derivative for speed.

Compare my reasoning

The quotient rule gives q′(t) = [10(t + 2) − 10t]/(t + 2)² = 20/(t + 2)².
The original distance at t = 3 is 30/5 = 6 metres.
The speed is 20/25 = 4/5 m/s; the evaluation time belongs to the supplied non-negative domain.

Distance 6 m; instantaneous speed 4/5 m/s at t = 3 s.

Look for these in your work

Simplify both quotient-rule contributions before evaluating.
Keep the quantity, derivative, units and model domain distinct.

Common mistakes

Reversing the quotient numerator order reverses its sign.
The denominator is squared in the derivative, but its original zeros stay excluded.
A positive derivative on two separate intervals does not justify comparing across an excluded pole.

Recall without your notes

What can you explain now?

Differentiate y = x²/(x + 1), keeping its excluded input.

Compare with the explanation

y′ = x(x + 2)/(x + 1)², with x ≠ −1.

The numerator is 2x(x + 1) − x² = x² + 2x; the original zero denominator remains excluded.

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Make it stick

Write the quotient rule and verify its sign using a simple linear-over-linear example. Explain why an original denominator exclusion survives differentiation and why separated domain intervals matter.

If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.

Keep learning Mathematics

Check the official syllabus 2026–2027

Course scope and assessment

312 sequenced lessons provide 936 guided, independent and application problems covering the 160 objectives in sections 1.1–1.8, 2.1–2.6, 3.1–3.9, 4.1–4.5, 5.1–5.5 and 6.1–6.5. The map lists all 38 sections across six alternative components; all eight Pure Mathematics 1 sections and all six Pure Mathematics 2 sections, from Paper 2 Algebra through Numerical solution of equations, currently have full teaching sequences. Paper 3 Algebra, section 3.1, links the six shared foundation lessons plus six lessons on partial fractions and rational binomial expansions; Paper 3 logarithms, trigonometry and numerical equations use the shared lessons, with the Paper 1 graph foundation linked into trigonometry. Paper 3 differentiation combines the shared methods with its inverse-tangent derivative lesson. Paper 3 Integration links four shared standard and identity lessons plus seven advanced lessons, covering all six section 3.5 requirements. Eight Vectors lessons cover all six section 3.7 requirements, with 2D and 3D line and solid applications. Nine Differential equations lessons cover all four section 3.8 requirements, including Paper 3 integration techniques, initial conditions, equilibrium solutions and conditional model interpretations. Eleven Complex numbers lessons cover all eight section 3.9 requirements, with seven Argand diagrams and explicit argument and boundary checks. All nine Pure Mathematics 3 sections now have teaching linked to every mapped requirement. Nine Forces and equilibrium lessons cover all seven section 4.1 requirements, with six force diagrams, signed incline-friction intervals, reaction-dependent pull and push limits and separate-body third-law checks. Nine Kinematics lessons cover the four official section 4.2 requirements, with the graph requirement mapped to three relationships. Six motion diagrams support directed quantities, graph slopes and signed areas, Paper 1 time-calculus methods, reversal distances, constant-acceleration stages and shared-clock particle meetings. Six Momentum lessons cover both section 4.3 requirements, including signed system totals, conservation assumptions, outgoing velocities, coalescence, unknown masses and successive impacts. Four diagrams show system momentum through the stipulated events, and supplied kinetic-energy checks distinguish conservation of momentum from changes in kinetic energy. Twelve Newton’s laws lessons cover all four section 4.4 requirements, with vertical support and release stages, rough horizontal and incline motion, inferred parameters, hanging and surface pulley systems, rigid tow-bar thrust and slack-string transitions. Seven force diagrams support the separate-particle balances. Nine Energy, work and power lessons cover all five section 4.5 requirements, with angle-dependent work, kinetic and reference potential energy, signed net-work balances, smooth curved slides, rough and driven slopes, ideal connected-system energy and instantaneous power on hills. Five diagrams distinguish the energy inputs, losses and valid positive-speed force model. All five Mechanics sections now have teaching and staged practice linked to their mapped requirements. Thirteen Representation of data lessons cover all five section 5.1 requirements, including choice and limitations of displays, ordered and back-to-back stems, raw summaries and box plots, unequal-width density histograms, cumulative-frequency interpolation, descriptive moment calculations, grouped estimates, coding, combining two datasets and correcting totals. Six accessible diagrams retain repeated observations and distinguish numerical spread, frequency area and estimated readings. Ten Permutations and combinations lessons cover both official section 5.2 requirements, with selection and ordered roles, repeated objects, adjacency blocks, non-adjacency gaps, leading-zero cases, labelled multi-row seating and mixed restrictions. Six accessible diagrams show outcome trees, ordered versus unordered choices, repeated copies, blocks, gaps and distinct row seats. Eleven Probability lessons cover all four section 5.3 requirements, with equiprobable elementary outcomes, selection-based probabilities, complements and disjoint regions, independence tests, restricted sample spaces, two-way tables, replacement and removal trees, reversed conditions and unknown parameters. Seven accessible diagrams show dice outcomes, random subsets, event regions, a conditioned grid, a count table and two probability trees. Twelve Discrete random variables lessons cover all three official section 5.4 requirements, with complete probability tables, moment and parameter checks, binomial assumptions, exact probabilities and moments, geometric first-success probabilities, tails, expectation and integer reliability thresholds. Six accessible diagrams distinguish outcome grouping, weighted moments, fixed trials, discrete probability bars, waiting conventions and integer tail deadlines. Twelve Normal distribution lessons cover all three official section 5.5 requirements, with continuous-area interpretation, MF19 cumulative tables, full standardisation, tail and interval regions, conditional probabilities, percentiles, one- and two-parameter inference, binomial suitability, continuity corrections and approximate integer thresholds. Seven accessible diagrams show model areas, table reading, aligned scales, intervals, quantiles, matched binomial and normal shapes and whole integer cells. All five Statistics 1 sections now have teaching and staged practice linked to their mapped requirements. Fourteen Poisson distribution lessons cover all five official section 6.1 requirements, with homogeneous event-count assumptions, exposure scaling, exact masses and tails, conditional count events, moments, parameter inference, probability duration targets, disjoint and overlapping windows, rare-count binomial approximations, matched normal approximations and continuity corrections. Nine accessible diagrams show event exposures, unit conversions, discrete bars, complete tail complements, population moments, shared windows, approximate count models and integer cells. Exact checks distinguish approximate and exact close-target decisions. Fourteen Linear combinations lessons cover the six listed results in official section 6.2, with affine moments, signed scales and constant cases, expectation without independence, independent squared variance contributions, fixed and random fees, independent copies versus reused values, affine normal events, independent normal sums and differences, weighted and conditional areas, coefficient inference, exact independent Poisson sums and overlapping exposure limits. Nine accessible diagrams distinguish scale changes, variance contributions, component costs, copy mechanisms, aligned normal scales, summed areas, Poisson count pairs and unsupported distribution claims. Fourteen Continuous random variables lessons cover both official section 6.3 requirements, with density validity and units, support-aware intervals, complements and conditional regions, first and second moments, percentiles by direct density areas, symmetry checks, parameter inference, infinite-support power and exponential integrals, divergent-moment limits and combined applications. Nine accessible diagrams distinguish density height from probability, normalisation, negative-density failures, central and conditional areas, mean versus median, percentile locations and partial infinite tails. Explicit cumulative distribution function knowledge is not required. Eighteen Sampling and estimation lessons cover all eight official section 6.4 requirements, with population and frame definitions, random-number selection, sampling-quality limits, independent sample-mean moments, normal means, CLT conditions, SE scaling, raw, summarised and coded unbiased estimates, known-variance and large-sample mean intervals, repeated-sampling interpretation, precision planning, proportion intervals, sparse-outcome limits and contextual applications. Ten accessible diagrams distinguish selection frames, number rejection, repeated sample means, population versus mean density, CLT models, variance divisors, interval targets, illustrative coverage, integer size checks and proportion margins. Eighteen Hypothesis tests lessons cover all five official section 6.5 requirements, with pre-specified alternatives, inclusive count tails, explicit equal-tail conventions, integer critical regions and actual significance, exposure-aware Poisson tests, continuity-corrected count approximations, normal and large-sample mean tests, confidence-interval connections, error classifications, fixed-region binomial, Poisson and normal error probabilities and sample-size power comparisons. Nine accessible diagrams show the testing procedure, exact count regions, unequal Poisson tails, discrete significance jumps, corrected count boundaries, matching mean intervals and alternative error distributions. All shared lessons are counted once in the course total. All 38 sections now link teaching and practice. Eight earlier overview lessons remain available, but do not count as full section coverage. This is a content-availability measure, not full programme completion or exam readiness. Written practice is self-checked, not officially marked. Check the 2028–2030 syllabus separately if sitting an examination from 2028; this teaching map uses 2026–2027 requirements.