A Level · Cambridge (CIE) · 9709

Differentiate trigonometric composites using radian measure

Use the sine, cosine and tangent derivatives with their inner factors, signs and original tangent exclusions.

Section 3.4 · 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

Use sine, cosine, tangent and secant and understand radian measure.
Apply the chain rule and evaluate exact trigonometric values.

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

Quick readiness check

What is d(cos x)/dx for radian x?

What you will learn

Differentiate sin x, cos x and tan x in radians and apply the chain rule.
Connect a derivative's value to a curve gradient or a signed motion rate, keeping domains and units.

The solid blue graph is the original height y = sin x; the dashed red graph is its gradient function y′ = cos x, with x in radians. At x = π/2, the original height is 1 but the gradient is 0, so the original curve has a horizontal tangent. At x = π/6, the height is 1/2 and the gradient √3/2. The two graphs describe different quantities. This window shows one complete turn; both functions continue periodically.

The standard trigonometric derivatives assume radians

With radian input x, d(sin x)/dx = cos x, d(cos x)/dx = −sin x and d(tan x)/dx = sec² x. Tangent and its derivative require cos x ≠ 0. For an inner rule u(x), multiply by u′: the derivatives are u′ cos u, −u′ sin u and u′ sec² u, with the tangent domain cos u ≠ 0. Keep a cosine derivative's negative sign separate from any outside coefficient.

These formulae are for radians. If a function is defined using a degree variable d, write its radian input explicitly: sin(πd/180) differentiates with respect to d to (π/180)cos(πd/180). A calculator's display mode does not remove this scale factor. In the usual syllabus calculus questions, use the radian input and retain every stated inner factor.

A curve height and its gradient are different outputs

For y = 3 sin 2x − 4 cos x + tan 3x, differentiation gives y′ = 6 cos 2x + 4 sin x + 3 sec² 3x, where cos 3x ≠ 0. The sign of the second term is positive because the outside −4 multiplies the cosine derivative's negative sign. For y = sin x, at x = π/2 the curve height is 1 but its derivative cos x is 0. At x = π/6 the height is 1/2 and the gradient is √3/2.

A supplied signed displacement S(t) = 4 sin(πt/6) cm, with t in seconds, has velocity S′(t) = (2π/3)cos(πt/6) cm/s. At t = 3 seconds the displacement is 4 cm and the velocity is zero; at t = 6 seconds the velocity is −2π/3 cm/s. A negative signed velocity indicates decreasing displacement, not a negative speed. The phase coefficient π/6 is an inner rate measured in radians per second.

Pause and explain

What is the derivative of tan(x²) where the original function is defined?

Put the idea to work

Worked example

Differentiate y = 3 sin 2x − 4 cos x + tan 3x, using radians and stating the original tangent restriction.

Show the worked solution

The sine term gives 3(2)cos 2x = 6 cos 2x.
The cosine term gives −4(−sin x) = 4 sin x; the tangent composite gives 3 sec² 3x.
Add the terms and retain cos 3x ≠ 0, the original tangent's denominator restriction.

Answer y′ = 6 cos 2x + 4 sin x + 3 sec² 3x, with cos 3x ≠ 0.

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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 = 2 cos 3x + sin x in radians and evaluate y′ at x = 0.

Give me a hint

Differentiate the cosine composite with both its negative sign and factor 3.

Compare my reasoning

The first term gives −6 sin 3x and the second gives cos x.
Thus y′ = −6 sin 3x + cos x for every real x.
At zero, sine is 0 and cosine is 1, so y′(0) = 1.

y′ = −6 sin 3x + cos x; y′(0) = 1.

Look for these in your work

Keep the composite factor and cosine derivative sign.
Evaluate the derivative rather than the original curve height.

2 · Independent

Differentiate y = tan(x²) in radians and state the restriction on its finite derivative.

Give me a hint

Use the tangent chain derivative and check its cosine denominator.

Compare my reasoning

Set u = x², with inner derivative 2x.
The outer derivative is sec² u, so y′ = 2x sec²(x²).
Require cos(x²) ≠ 0; this preserves the original tangent domain, including valid input x = 0.

y′ = 2x sec²(x²), with cos(x²) ≠ 0.

Look for these in your work

Retain the unchanged composite argument.
State the original tangent's excluded inputs through its cosine condition.

3 · Transfer

A supplied signed displacement is S(t) = 4 sin(πt/6) cm, with t in seconds. Find the velocity at t = 3 and 6 seconds and interpret the sign.

Give me a hint

Differentiate with respect to time and retain the phase's inner rate π/6.

Compare my reasoning

The velocity is S′(t) = (2π/3)cos(πt/6) cm/s.
At t = 3 the cosine is zero, giving 0 cm/s; at t = 6 it is −1, giving −2π/3 cm/s.
The negative value means displacement is decreasing at that instant; its speed is the positive magnitude 2π/3 cm/s.

Velocity 0 cm/s at t = 3; −2π/3 cm/s at t = 6; speed at t = 6 is 2π/3 cm/s.

Look for these in your work

Use the phase's radian rate in the chain rule.
Distinguish signed velocity from its non-negative speed magnitude.

Common mistakes

The standard derivatives use radians, not an unconverted degree variable.
A cosine derivative contributes a negative sign.
Keep the inner rate and the original tangent denominator restriction.

Recall without your notes

What can you explain now?

Give the height and gradient of y = sin x at x = π/2.

Compare with the explanation

Height 1; gradient 0.

Evaluate sine for the height and its derivative cosine for the gradient.

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Make it stick

State the three trigonometric derivative rules and their angle-unit assumption. Explain how to differentiate a composite tangent and distinguish a sine height from its cosine gradient.

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.