A Level · Cambridge (CIE) · 9709

Build tangent and normal equations at a curve point

Evaluate the curve and its derivative at the same input, use a point-gradient line and handle a horizontal tangent's vertical normal.

Section 1.7 · Differentiation

Pure Mathematics 1 · Differentiation · 1.7

Pure Mathematics 1 (Paper 1) · AS & A Level Mathematics 9709, 2026–2027. Use an informal understanding of limits; formal general differentiation from first principles and points of inflexion are not required in section 1.7. Written practice is self-checked.

Before you begin

Differentiate powers and composite functions.
Use point-gradient line equations and perpendicular gradients.

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

Quick readiness check

For y = x² + 1 at x = 1, what is the curve point?

What you will learn

Find tangent and normal equations using a differentiated curve.
Handle zero tangent gradients without dividing by zero.

The point comes from the curve; the gradient comes from its derivative

At input x = a on y = f(x), the curve point is (a, f(a)) and its finite tangent gradient is f′(a). The tangent equation is y − f(a) = f′(a)(x − a). Substituting a into the derivative gives a gradient, not the point's height. Find both quantities before building the line.

For y = x² − 3x + 2 at x = 3, f(3) = 2 and f′(3) = 2(3) − 3 = 3. The tangent is y − 2 = 3(x − 3), or y = 3x − 7. It passes through (3, 2) and its coefficient of x is the calculated gradient 3; these two checks distinguish it from a line with the correct gradient but wrong intercept.

A normal is perpendicular to the tangent

When the tangent gradient m is finite and non-zero, the normal has gradient −1/m through the same curve point. For the example, the normal is y − 2 = −(x − 3)/3, equivalently x + 3y = 9. The product of the gradients is −1 and both lines contain the point.

If m = 0, the tangent is horizontal and the normal is vertical. For y = (x − 2)⁴ + 1 at x = 2, the point is (2, 1), the tangent is y = 1 and the normal is x = 2. Do not try to turn −1/0 into a finite gradient or force the normal into y = mx + c. Where a finite derivative does not exist, the usual point-gradient procedure needs separate geometric consideration.

Pause and explain

A curve's finite tangent gradient is zero at (2, 1). What is its normal equation?

Put the idea to work

Worked example

Find the tangent and normal to y = x² − 3x + 2 at x = 3.

Show the worked solution

Evaluate the curve to get (3, 2), and differentiate to get y′ = 2x − 3, hence tangent gradient 3.
The tangent is y − 2 = 3(x − 3), or y = 3x − 7.
The normal gradient is −1/3, so y − 2 = −(x − 3)/3, or x + 3y = 9; check both at (3, 2).

Answer Tangent y = 3x − 7; normal x + 3y = 9.

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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

Find tangent and normal equations to y = x² + 1 at x = 1.

Give me a hint

Find the point and tangent gradient separately, then use the same point for both lines.

Compare my reasoning

The point is (1, 2), and y′ = 2x gives tangent gradient 2.
The tangent is y − 2 = 2(x − 1), hence y = 2x.
The normal gradient is −1/2, so y − 2 = −(x − 1)/2, giving y = −x/2 + 5/2.

Tangent y = 2x; normal y = −x/2 + 5/2.

Look for these in your work

Use the original curve and derivative for their different roles.
Verify both lines pass through the same point and have perpendicular gradients.

2 · Independent

Find the tangent and normal to y = √(4x + 1) at x = 2.

Give me a hint

Use the chain rule for the gradient and verify the input lies inside the finite-derivative domain.

Compare my reasoning

The point is (2, 3), and y′ = 2/√(4x + 1) gives gradient 2/3.
The tangent y − 3 = (2/3)(x − 2) simplifies to y = 2x/3 + 5/3.
The normal gradient is −3/2, so its equation is y − 3 = −3(x − 2)/2, or y = −3x/2 + 6.

Tangent y = 2x/3 + 5/3; normal y = −3x/2 + 6.

Look for these in your work

Retain the inner derivative factor in the root calculation.
Use a negative reciprocal and the actual curve point.

3 · Transfer

For y = (x − 2)⁴ + 1, a learner tries to use −1/0 as a normal gradient at x = 2. Give both correct line equations and explain the geometric correction.

Give me a hint

Differentiate the curve and identify the perpendicular line from its direction.

Compare my reasoning

The curve point is (2, 1), and y′ = 4(x − 2)³ gives tangent gradient zero.
The horizontal tangent through the point is y = 1.
Its perpendicular normal is vertical, x = 2, with undefined finite gradient; it should not be represented by division by zero.

Tangent y = 1; normal x = 2; use perpendicular directions rather than −1/0.

Look for these in your work

Identify the zero-gradient exception correctly.
Write the vertical normal in an appropriate line form.

Common mistakes

Evaluate the original curve for the point's height, not the derivative.
Use the negative reciprocal for a non-zero tangent gradient.
A horizontal tangent has a vertical normal with an x = constant equation.

Recall without your notes

What can you explain now?

At the point (2, 1) on y = (x − 2)⁴ + 1, find tangent and normal equations.

Compare with the explanation

Tangent y = 1; normal x = 2.

The derivative is 4(x − 2)³, so it is zero at the point; perpendicularity therefore gives a vertical normal.

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Make it stick

Explain why the original function and its derivative are both needed to build a tangent. Then describe how the normal changes when the tangent gradient is zero.

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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.