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Quick differentiation map
Core Derivative Rules
Trigonometric and Hyperbolic Functions
Implicit Differentiation
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.
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.
Quick differentiation map
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First action
1 second
Differentiation measures rate of change.
Identify the variable.
10 seconds
The form decides the rule.
Choose chain, product, quotient, implicit, or parametric.
100 seconds
Applications need interpretation.
Link the derivative to tangent, normal, maximum, minimum, or rate.
, the derivative is zero, so the tangent is horizontal.
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).
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.
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.
: 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".
