H2 Maths Differentiation Formula Sheet | Rules & Techniques
H2 Maths Differentiation Formula Sheet | Rules & Techniques
Study guide/
H2 Maths differentiation formula sheet: standard derivatives, product/quotient/chain rules, implicit and parametric differentiation, stationary points, and related rates — align...
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−x, use the product rule first. At x=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 signal
Rule to choose first
Common trap
A function inside another function, such as sin((3x + 1)^2)
Chain rule
Differentiating the outside but forgetting the inside derivative
Two changing factors multiplied, such as x^2 ln x
Product rule
Differentiating only one factor
A changing numerator over a changing denominator
Quotient rule
Reversing u'v - uv' or differentiating the fraction term by term
An equation mixing x and y, such as x^2 + xy + y^2 = 7
Implicit differentiation
Forgetting that differentiating a y term introduces dy/dx
Both x and y written in terms of t
Parametric differentiation
Reporting 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
Function
Derivative
xn
nxn−1
ekx
kekx
lnx
x1
sinx
cosx
cosx
−sinx
tanx
sec2x
sin−1x
1−x21
sinhx
coshx
coshx
sinhx
Rules
Rule
Formula
Product
(uv)′=u′v+uv′
Quotient
(vu)′=v2u′v−uv′
Chain
dxdf(g(x))=f′(g(x)),g′(x)
Implicit & parametric
Form
Derivative
Implicit
Differentiate both sides in x, then solve for dxdy
Parametric
dxdy=dx/dtdy/dt
Applications
Quantity
Formula
Tangent gradient
mt=dxdy
Normal gradient
mn=−mt1
Stationary points
dxdy=0; minimum if dx2d2y>0
Related rates
dtdz=dxdzdtdx
Core Derivative Rules
Power rule: dxdxn=nxn−1 for rational n.
Product rule: dxd[uv]=u′v+uv′
Quotient rule: dxd(vu)=v2u′v−uv′
Chain rule: dxdf(g(x))=f′(g(x))g′(x)
Exponential and logarithmic derivatives: dxdekx=kekx, dxdlnx=x1
Trigonometric and Hyperbolic Functions
dxdsinx=cosx, dxdcosx=−sinx, dxdtanx=sec2x.
Inverses: dxdsin−1x=1−x21
Hyperbolic: dxdsinhx=coshx, dxdcoshx=sinhx
Implicit Differentiation
Differentiate both sides treating y as function of x and solve for dxdy.
Example -- Implicit derivative
Given x2+xy+y2=7.
Differentiate: 2x+y+xdxdy+2ydxdy=0.
Factor: (x+2y)dxdy=−(2x+y).
Hence dxdy=−x+2y2x+y
Parametric Differentiation
For x=f(t), y=g(t): dxdy=dtdxdtdy.
Second derivative: dx2d2y=dtdx1dtd(dxdy)
Example -- Parametric slope
Given x=t2+1, y=ln(1+t).
dtdx=2tdtdy=1+t1.
dxdy=2t(1+t)1.
Tangents and Normals
Tangent gradient mt=dxdy at point; normal gradient mn=−mt1.
Tangent line: y−y0=mt(x−x0)
Normal line: y−y0=mn(x−x0)
Example -- Tangent to curve
Find tangent at x=1 for y=xe−x.
dxdy=e−x−xe−x=e−x(1−x).
At x=1, gradient zero → tangent horizontal: y=e1.
Optimisation and Stationary Points
Stationary points satisfy dxdy=0; classify with second derivative test.
Second derivative: dx2d2y>0 (minimum), <0 (maximum). If dx2d2y(x0)=0, differentiate repeatedly until the first non-zero dxmdmy(x0) appears - an even-order derivative reveals a minimum (positive) or maximum (negative), while an odd-order derivative signals a point of inflection. Alternatively, chart the sign of dxdy on either side of x0 to confirm behaviour.
For optimisation, interpret result in context (length, area, cost).
Example -- Optimisation
Minimise surface area of open-top box with base square of side x and volume 108cm3.
Height h=x2108.
Surface area S=x2+4xh=x2+x432.
Differentiate: dxdS=2x−x2432=0⇒2x3=432
x=3216=6, h=3
Related Rates
Differentiating implicit relationships with respect to time: dtdz=dxdzdtdx.
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 cue
First setup
Differentiation move
Common trap
Area or volume changing
Write the area or volume formula using the changing length.
Differentiate both sides with respect to t.
Substituting the current length before differentiating.
Two lengths linked by geometry
Use Pythagoras, similar triangles, or a trigonometric relation.
Differentiate the relationship implicitly with respect to t.
Treating the other length as constant when it is also changing.
Speed given in one direction
Decide whether the rate is positive or negative for the chosen variable.
Attach the sign to dtdx or dtdr.
Using speed as a positive number even when the distance is decreasing.
Asked for units of the final rate
Track the units of the target quantity.
Quote area rate, volume rate, or length rate with matching units.
Giving cm⋅s−1 for an area rate.
Worked check: if the area of a circle is A=πr2 and dtdr=0.2cm⋅s−1, differentiate first to get dtdA=2πrdtdr. Only then substitute r=5cm, giving dtdA=2πcm2⋅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 grows at dtdr=0.2cm⋅s−1. Find rate of change of area when r=5cm.
Area A=πr2.
dtdA=2πrdtdr=2π×5×0.2=2πcm2⋅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 dxdy 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 is m-times differentiable in a neighbourhood of x0 and that f′(x0)=f′′(x0)=⋯=f(m−1)(x0)=0 while f(m)(x0)=0. Taylor's theorem with remainder gives
f(x)=f(x0)+m!f(m)(x0)(x−x0)m+Rm(x),
where the remainder satisfies ∣Rm(x)∣≤K∣x−x0∣m+1 for some constant K when x is close to x0. For sufficiently small ∣x−x0∣, the dominant term is therefore m!f(m)(x0)(x−x0)m.
If m is even, (x−x0)m≥0 on both sides of x0. A positive coefficient f(m)(x0) forces f(x)≥f(x0) nearby (local minimum), while a negative coefficient gives f(x)≤f(x0) (local maximum).
If m is odd, the sign of (x−x0)m flips across x0
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) or ex2, 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: dxdcosx=−sinx - the negative sign is mandatory. A common slip is writing +sinx, especially after a chain-rule step where sign tracking becomes cluttered.
Confusing dxdy with dx2d2y
Failing to use implicit differentiation when y cannot be isolated: On curves such as x3+y3=6xy, attempting to rearrange for y
Missing the product rule when two functions multiply: Expressions like x2lnx or exsinx require the product rule (uv)′=u′v+uv′
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 or ∞∞ 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: dx2d2y>0 gives a minimum and <0 gives a maximum - but it is inconclusive when the second derivative equals zero. The sign-change test (checking the sign of dxdy on either side of the stationary point) always gives a definitive classification. For A-level questions where the second derivative is messy or evaluates to zero at the stationary point, the sign-change test is the safer choice.
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:
: Setting the second derivative to zero and concluding "minimum" without further analysis loses the method mark. If dx2d2y=0, you must apply the higher-order test or a sign-change chart for dxdy.
before differentiating wastes time and is often impossible. Differentiate term by term directly and solve for
dxdy
.
. A frequent error is differentiating only one factor - typically seen when the second factor looks "simple".
