H2 Maths Differentiation Formula Sheet | Rules & Techniques

Study guideUpdated 17 Jul 2026

H2 Maths differentiation formula sheet: standard derivatives, product/quotient/chain rules, implicit and parametric differentiation, stationary points, and related rates — align...

Q: What does H2 Maths Notes (JC 1-2): 5.1) Differentiation cover?
A: Differentiation rules, implicit methods, and optimisation for H2 Maths Topic 5.1.
Download: Get the H2 Maths Differentiation formula sheet (PDF) for quick revision, or the complete notes (PDF) for the full walkthrough.
Before you revise
Memorise derivative rules with the full statement (chain, product, quotient). Practise switching between coordinate, parametric, and implicit forms so you can differentiate anything the paper throws at you.
  • Differentiation measures rate of change: Identify the variable.
  • The form decides the rule: Choose chain, product, quotient, implicit, or parametric.
  • Applications need interpretation: Link the derivative to tangent, normal, maximum, minimum, or rate.

Concrete example: For y=xe−xy=xe^{-x}, use the product rule first. At x=1x=1, the derivative is zero, so the tangent is horizontal.

Before differentiating, choose the structure of the expression first. Most wrong H2 differentiation solutions start with the right derivative facts but the wrong rule.

Expression signalRule to choose firstCommon trap
A function inside another function, such as sin((3x + 1)^2)Chain ruleDifferentiating the outside but forgetting the inside derivative
Two changing factors multiplied, such as x^2 ln xProduct ruleDifferentiating only one factor
A changing numerator over a changing denominatorQuotient ruleReversing u'v - uv' or differentiating the fraction term by term
An equation mixing x and y, such as x^2 + xy + y^2 = 7Implicit differentiationForgetting that differentiating a y term introduces dy/dx
Both x and y written in terms of tParametric differentiationReporting dy/dt instead of dy/dx

Status: SEAB's current H2 Mathematics (9758) syllabus PDF is labelled for 2026. Topic 5.1 scope is within Section A Pure Mathematics, which is assessed in Paper 1 (100 marks) and Paper 2 Section A (40 marks).

Formulas at a glance

Every rule and standard derivative the 9758 syllabus expects you to apply, on one screen. The differentiation rules (product, quotient, chain) are not given in MF27, so memorise them. Worked examples for each appear in the sections below.

Standard derivatives

FunctionDerivative
xnx^nnxn−1nx^{n-1}
ekxe^{kx}

Rules

RuleFormula
Product(uv)′=u′v+uv′(uv)' = u'v + uv'

Implicit & parametric

FormDerivative
ImplicitDifferentiate both sides in xx, then solve for dydx\dfrac{\mathrm{d}y}{\mathrm{d}x}
Parametricdydx=dy/dtdx/dt\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{\mathrm{d}y/\mathrm{d}t}{\mathrm{d}x/\mathrm{d}t}

Applications

QuantityFormula
Tangent gradientmt=dydxm_t = \dfrac{\mathrm{d}y}{\mathrm{d}x}
Normal gradientmn=−1mtm_n = -\dfrac{1}{m_t}

Core Derivative Rules

  • Power rule: ddxxn=nxn−1 \dfrac{\mathrm{d}}{\mathrm{d}x} x^n = nx^{n-1} for rational n n

Trigonometric and Hyperbolic Functions

  • ddxsin⁡x=cos⁡x \dfrac{\mathrm{d}}{\mathrm{d}x} \sin x = \cos x , ddxcos⁡x=−sin⁡x \dfrac{\mathrm{d}}{\mathrm{d}x} \cos x = -\sin x

Implicit Differentiation

  • Differentiate both sides treating y y as function of x x and solve for dydx \dfrac{\mathrm{d}y}{\mathrm{d}x} .

Example -- Implicit derivative

Given x2+xy+y2=7 x^2 + xy + y^2 = 7 .

  1. Differentiate: 2x+y+xdydx+2ydydx=0 2x + y + x \dfrac{\mathrm{d}y}{\mathrm{d}x} + 2y \dfrac{\mathrm{d}y}{\mathrm{d}x} = 0

Parametric Differentiation

  • For x=f(t) x = f(t) , y=g(t) y = g(t) : dydx=dydtdxdt \dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{\dfrac{\mathrm{d}y}{dt}}{\dfrac{\mathrm{d}x}{dt}}

Example -- Parametric slope

Given x=t2+1 x = t^2 + 1 , y=ln⁡(1+t) y = \ln(1 + t) .

  1. dxdt=2t dydt=11+t \dfrac{\mathrm{d}x}{dt} = 2t \, \dfrac{\mathrm{d}y}{dt} = \dfrac{1}{1 + t}

Tangents and Normals

A curve with a local maximum and a local minimum. At each stationary point the tangent is horizontal, so dy/dx = 0. Elsewhere the gradient of the curve equals the gradient of the tangent line at that point.xydy/dx = 0dy/dx = 0tangent
The gradient of a curve at a point is the gradient of its tangent there, given by dy/dx. At a stationary point the tangent is horizontal, so dy/dx = 0; the second derivative (or a sign test on dy/dx) then classifies it as a maximum, minimum, or point of inflexion.
  • Tangent gradient mt=dydx m_t = \dfrac{\mathrm{d}y}{\mathrm{d}x} at point; normal gradient mn=−1mt m_n = -\dfrac{1}{m_t}

Example -- Tangent to curve

Find tangent at x=1 x = 1 for y=xe−x y = x e^{-x} .

  1. dydx=e−x−xe−x=e−x(1−x) \dfrac{\mathrm{d}y}{\mathrm{d}x} = e^{-x} - x e^{-x} = e^{-x}(1 - x)

Optimisation and Stationary Points

  • Stationary points satisfy dydx=0 \dfrac{\mathrm{d}y}{\mathrm{d}x} = 0 ; classify with second derivative test.
  • Second derivative: d2ydx2>0 \dfrac{\mathrm{d}^2 y}{\mathrm{d}x^2} > 0

Example -- Optimisation

Minimise surface area of open-top box with base square of side x x and volume 108cm3 108 \pu{cm3} .

  1. Height h=108x2 h = \dfrac{108}{x^2} .
  2. Surface area S=x2+4xh=x2+432x S = x^2 + 4xh = x^2 + \dfrac{432}{x}

Related Rates

  • Differentiating implicit relationships with respect to time: dzdt=dzdxdxdt \dfrac{dz}{dt} = \dfrac{dz}{\mathrm{d}x} \dfrac{\mathrm{d}x}{dt}

Related-rate setup checkpoint

Before differentiating, write the static relationship between the quantities first. The time derivative comes after the geometry or formula is correct.

Question cueFirst setupDifferentiation moveCommon trap
Area or volume changingWrite the area or volume formula using the changing length.Differentiate both sides with respect to tt.Substituting the current length before differentiating.
Two lengths linked by geometryUse Pythagoras, similar triangles, or a trigonometric relation.Differentiate the relationship implicitly with respect to tt.Treating the other length as constant when it is also changing.
Speed given in one directionDecide whether the rate is positive or negative for the chosen variable.Attach the sign to

Worked check: if the area of a circle is A=πr2A=\pi r^2 and drdt=0.2 cm ⋅ s−1\dfrac{\mathrm{d}r}{\mathrm{d}t}=\pu{0.2 cm.s-1}

Misconception check: related-rate questions are not solved by plugging values into the original formula first. Differentiate the relationship while the variables are still variables, then substitute the instant described in the question.

Example -- Expanding circle

A circle radius r r grows at drdt=0.2cm ⋅ s−1 \dfrac{dr}{dt} = 0.2 \pu{cm.s-1}

  1. Area A=πr2 A = \pi r^2 .
  2. dAdt=2πrdrdt=2π×5×0.2=2πcm2 ⋅ s−1 \dfrac{dA}{dt} = 2\pi r \dfrac{dr}{dt} = 2\pi \times 5 \times 0.2 = 2\pi \pu{cm2.s-1}

Calculator Workflow

  • The graphing calculator (GC) differentiation function evaluates derivatives numerically; record inputs (e.g. d/dx at specific points).
  • Use TABLE to evaluate derivative sign around stationary points for classification.
  • Store intermediate expressions to avoid algebra slips when differentiating complex functions.

Exam Watch Points

  • Present exact derivatives before substituting numerical values.
  • State the method used (implicit, parametric) and justify each step clearly.
  • Include units in related-rates answers.
  • For optimisation, verify solutions satisfy constraints (positive dimensions, etc.).

Practice Quiz

Test differentiation fluency, stationary point analysis, and related-rate workflows under exam-style pressure.

Quick Revision Checklist

  • Apply product, quotient, and chain rules without hesitation.
  • Differentiate implicit and parametric relations, reporting dydx \dfrac{\mathrm{d}y}{\mathrm{d}x} clearly.
  • Evaluate tangents, normals, and stationary points with proper classification.
  • Tackle optimisation and related-rate problems with a structured plan and unit-aware answers.

Appendix: Higher-Order Derivative Test (Proof Sketch)

Suppose f f is m m -times differentiable in a neighbourhood of x0 x_0 and that f′(x0)=f′′(x0)=⋯=f(m−1)(x0)=0 f'(x_0) = f''(x_0) = \cdots = f^{(m-1)}(x_0) = 0

f(x)=f(x0)+f(m)(x0)m!(x−x0)m+Rm(x), f(x) = f(x_0) + \dfrac{f^{(m)}(x_0)}{m!} (x - x_0)^m + R_m(x),

where the remainder satisfies ∣Rm(x)∣≤K∣x−x0∣m+1 \lvert R_m(x) \rvert \leq K \lvert x - x_0 \rvert^{m+1}

  • If m m is even, (x−x0)m≥0 (x - x_0)^m \geq 0 on both sides of x0 x_0

This argument justifies the higher-order derivative test and matches the sign-chart alternative described earlier.

Want weekly guided practice on Differentiation? Our H2 Maths tuition programme builds fluency in this topic through structured problem sets and exam-style drills.

Common exam mistakes

  • Forgetting the chain rule on composite functions: When differentiating expressions such as sin⁡(3x2+1) \sin(3x^2 + 1) or ex2 e^{x^2} , the outer derivative must be multiplied by the inner derivative. Omitting the inner factor is the single most penalised slip in Paper 1 differentiation questions.
  • Sign error on the derivative of cosine: ddxcos⁡x=−sin⁡x \dfrac{\mathrm{d}}{\mathrm{d}x} \cos x = -\sin x

Frequently asked questions

Which paper does Differentiation (Topic 5.1) appear in?
Differentiation is a Pure Mathematics topic and is examined in both Paper 1 (100 marks, pure only) and Paper 2 Section A (40 marks, pure). [1] Related-rate and optimisation questions can appear in either paper, so you should expect differentiation to account for a significant portion of the pure marks across both sittings.

Is L'Hôpital's rule in the H2 Maths 9758 syllabus?
No. L'Hôpital's rule is not listed in the SEAB 9758 syllabus and will not be credited in an A-level answer. [1] Limits involving 00 \dfrac{0}{0} or ∞∞ \dfrac{\infty}{\infty} indeterminate forms are typically approached via Maclaurin series (Topic 5.2) or algebraic simplification. Using L'Hôpital's rule risks scoring zero for that part even if the numerical answer is correct.

Should I use the second derivative test or the sign-change test to classify stationary points?
Both are accepted, but each has a failure mode. The second derivative test is faster: d2ydx2>0 \dfrac{\mathrm{d}^2 y}{\mathrm{d}x^2} > 0 gives a minimum and <0 < 0

Is there a formula sheet for H2 Maths differentiation?
Yes - the "Formulas at a glance" section near the top of this page collects every result you need: the standard derivatives, the product, quotient, and chain rules, and the implicit, parametric, tangent/normal, and related-rate setups. The differentiation rules themselves are not given in MF27, so they must be memorised.

Other H2 Maths formula sheets

Revising more than one topic? Grab the matching one-page formula sheet:

For the official SEAB reference booklet, see the H2 Maths MF27 formula list.

Sources

  • SEAB H2 Mathematics syllabus (9758), examinations from 2026 - Topic 5 Calculus sub-topic 5.1 Differentiation (rules, parametric/implicit forms, tangents/normals, optimisation, related rates): SEAB 9758 syllabus PDF

Next steps: Stay aligned with the H2 Maths notes hub and continue with Topic 5.2 - Maclaurin Series.

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Marcus Pang
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Sources

  1. SEAB H2 Mathematics (9758) Syllabus 2026