A Level · Cambridge (CIE) · 9709

Differentiate implicit curves using product and chain rules

Treat y as changing with x, collect every dy/dx term and use the supplied curve point to choose a local gradient.

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

Use product and chain rules with a changing inner variable.
Collect a common derivative factor and build tangent and normal lines.

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

Quick readiness check

What is d(y²)/dx when y depends on x?

What you will learn

Find a first derivative from an implicit relation, including powers, products and natural-function terms in y.
Verify the supplied point and find tangent and normal lines on its local branch.

The blue implicit curve is x² + y² = xy + 7. P(1, 3) lies on it and has gradient (y − 2x)/(2y − x) = 1/5. The red tangent is x − 5y + 14 = 0; the dashed normal is 5x + y = 8. At x = 1, the other branch point (1, −2) instead has gradient 4/5. Both coordinates are needed to choose a local gradient. The axes have equal unit lengths. The full closed curve is drawn; the tangent and normal continue beyond the window.

Every occurrence of y carries its rate of change

An implicit equation relates x and y without necessarily solving for one global function y = f(x). On a differentiable local branch, treat y as a function of x. Thus d(y²)/dx = 2y(dy/dx) and d(eʸ)/dx = eʸ(dy/dx) by the chain rule. A product xy contributes y + x(dy/dx) because both its factors vary with x. Treating y as a constant would delete its changing contribution.

For x² + y² = xy + 7, differentiation gives 2x + 2y y′ = y + x y′. Collect to (2y − x)y′ = y − 2x, hence y′ = (y − 2x)/(2y − x) where 2y − x ≠ 0. Verify that the chosen point belongs to the original curve before substituting it. At (1, 3), 1 + 9 = 3 + 7 and the gradient is (3 − 2)/(6 − 1) = 1/5.

Choose the local branch using both coordinates

The tangent at (1, 3) is y − 3 = (x − 1)/5, or x − 5y + 14 = 0. The normal gradient is −5, giving 5x + y = 8. At the same x = 1, the original curve also includes (1, −2), which has a different gradient 4/5. A derivative expression involving both coordinates needs the supplied point, not x alone, to choose the local branch. A zero denominator requires a separate tangent check rather than an automatic finite gradient.

For eʸ + x² = e + 1, differentiation gives eʸ y′ + 2x = 0, so y′ = −2x/eʸ. At (1, 1) the gradient is −2/e. Eliminating y gives y = ln(e + 1 − x²) where |x| < √(e + 1), and its chain derivative agrees. This check preserves the original positive exponential quantity; inputs where e + 1 − x² is non-positive cannot lie on that real curve.

Pause and explain

On x² + y² = xy + 7, what must be used to evaluate a local gradient at x = 1?

Put the idea to work

Worked example

For x² + y² = xy + 7, find dy/dx and tangent and normal equations at (1, 3).

Show the worked solution

Differentiate to 2x + 2y y′ = y + x y′, using both chain and product rules.
Collect derivative terms to obtain y′ = (y − 2x)/(2y − x); the point satisfies the original curve and has gradient 1/5.
Use the point-gradient lines: tangent x − 5y + 14 = 0 and normal 5x + y = 8.

Answer y′ = (y − 2x)/(2y − x), where 2y − x ≠ 0; at (1, 3), tangent x − 5y + 14 = 0 and normal 5x + y = 8.

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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

For x² + y² = 25, find the gradient, tangent and normal at (3, 4).

Give me a hint

Differentiate implicitly and verify the point before evaluating its local gradient.

Compare my reasoning

The point is on the circle because 9 + 16 = 25; differentiation gives 2x + 2y y′ = 0 and y′ = −x/y for y ≠ 0.
At (3, 4), the gradient is −3/4, so the tangent is y − 4 = −3(x − 3)/4, or 3x + 4y = 25.
The normal gradient is 4/3, giving 4x − 3y = 0 through the same point.

Gradient −3/4; tangent 3x + 4y = 25; normal 4x − 3y = 0.

Look for these in your work

Use the chain factor when differentiating y².
Verify the point and both perpendicular line equations.

2 · Independent

For eʸ + x² = e + 1, find tangent and normal equations at (1, 1), then check the derivative using an explicit logarithmic form.

Give me a hint

Use the exponential chain derivative eʸ y′ and retain the real curve's positive log argument after eliminating y.

Compare my reasoning

The point satisfies the original equation; differentiation gives eʸ y′ + 2x = 0, so the gradient there is −2/e.
The tangent is y − 1 = −(2/e)(x − 1); the normal is y − 1 = (e/2)(x − 1).
The explicit form is y = ln(e + 1 − x²) for |x| < √(e + 1); differentiating it gives −2x/(e + 1 − x²), agreeing with −2x/eʸ on the curve.

Tangent y − 1 = −(2/e)(x − 1); normal y − 1 = (e/2)(x − 1); explicit check y = ln(e + 1 − x²), |x| < √(e + 1).

Look for these in your work

Differentiate the changing y inside the exponential.
Check the derivative and real domain against the equivalent explicit function.

3 · Transfer

For x² + xy + y² = 7, find tangent and normal equations at (1, 2). Explain how the product term changes the derivative.

Give me a hint

The xy derivative contributes y + x y′; collect all derivative terms after differentiating the full relation.

Compare my reasoning

The point satisfies 1 + 2 + 4 = 7; differentiation gives 2x + y + x y′ + 2y y′ = 0.
Hence y′ = −(2x + y)/(x + 2y), giving −4/5 at (1, 2), where its denominator is 5.
The tangent is 4x + 5y = 14 and the normal is 5x − 4y = −3; both pass through the original point.

Gradient −4/5; tangent 4x + 5y = 14; normal 5x − 4y = −3.

Look for these in your work

Include both contributions from the changing xy product.
Collect the derivative coefficient and verify both lines at a valid point.

Common mistakes

A y power or composite needs its inner derivative dy/dx.
The derivative of xy has both y and x(dy/dx) terms.
Verify the original point and use both coordinates for the local branch; a derivative denominator must be checked.

Recall without your notes

What can you explain now?

What is d(xy)/dx when y depends on x?

Compare with the explanation

y + x(dy/dx).

The product rule differentiates x in one contribution and the changing y in the other.

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Make it stick

Differentiate an implicit relation with a y square, an xy product and an exponential in y. Explain why the original point and a non-zero derivative coefficient are needed before finding its local lines.

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.