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UEC Math Wiki

A free study-notes library organised by the UEC syllabus, covering SC04 / SC05 / SC06 / SC07. Each unit covers the core concepts, the key formulas, worked examples, and the mistakes to watch for.

SC04Mathematics25 units
U01Quadratic Equations
U02Polynomials
U03Rational Expressions
U04Irrational Expressions
U05Angles and Their Units
U06Trigonometric Ratios of Acute Angles
U07Trigonometric Functions of Any Angle
U08Solution of Triangles
U09Trigonometric Identities and Equations
U10Rectangular Coordinates
U11Straight Lines
U12Sequences and Series
U13Simultaneous Equations
U14Matrices and Determinants
U15Inequalities
U17Solid Geometry
U18Statistics
U19Permutations and Combinations
U21Probability
U22Functions
U23Indices and Logarithms
U25Differentiation
U26Applications of Differentiation
U27Indefinite Integration
U28Definite Integration and Applications
SC05Advanced Mathematics28 units
U01Quadratic Equations
U02Polynomials
U03Rational Expressions
U04Irrational Expressions
U05Angles and Their Units
U06Trigonometric Ratios of Acute Angles
U07Trigonometric Functions of Any Angle
U08Solution of Triangles
U09Trigonometric Identities and Equations
U10Rectangular Coordinates
U11Straight Lines
U12Sequences and Series
U13Simultaneous Equations
U14Matrices and Determinants
U15Inequalities
U16Circles
U17Solid Geometry and Latitude/Longitude
U18Statistics
U19Permutations and Combinations
U20Binomial Theorem
U21Probability
U22Functions
U23Indices and Logarithms
U24Limits
U25Differentiation
U26Applications of Differentiation
U27Indefinite Integration
U28Definite Integration and Applications
SC06Advanced Mathematics I
Unit structure in progress
SC07Advanced Mathematics II5 units
第 1 章DifferentiationLIVE
第 2 章Integration
第 3 章Sequences & Series
第 4 章Vectors
第 5 章Complex Numbers
Unit structure follows the Dong Zong UEC math syllabus · full notes rolling out unit by unit
── SC07 · UNIT 1

Differentiation

Difficulty: ★★★★☆ · Prerequisites: algebra, basic limits

📌 Core concept

Differentiation finds the rate of change of a function. The derivative f′(x)f'(x) is the instantaneous rate of change of f(x)f(x) at xx — geometrically, the slope of the tangent line.

📋 Key formulas

Basic differentiation rules
Power rule: ddx[xn]=nxn−1\frac{d}{dx}[x^n] = nx^{n-1}
Product rule: ddx[uv]=u′v+uv′\frac{d}{dx}[uv] = u'v + uv'
Quotient rule: ddx[uv]=u′v−uv′v2\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{u'v - uv'}{v^2}
Chain rule: dydx=dydu⋅dudx\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}
Standard derivatives
ddx[sin⁡x]=cos⁡x\frac{d}{dx}[\sin x] = \cos x
ddx[cos⁡x]=−sin⁡x\frac{d}{dx}[\cos x] = -\sin x
ddx[tan⁡x]=sec⁡2x\frac{d}{dx}[\tan x] = \sec^2 x
ddx[ex]=ex\frac{d}{dx}[e^x] = e^x
ddx[ln⁡x]=1x\frac{d}{dx}[\ln x] = \dfrac{1}{x}
ddx[ax]=axln⁡a\frac{d}{dx}[a^x] = a^x \ln a

✏️ Worked examples

Example 1 — Chain rule
Differentiate y=sin⁡(3x2+1)y = \sin(3x^2 + 1).
  1. Let u=3x2+1u = 3x^2 + 1, so y=sin⁡uy = \sin u.
  2. Differentiate each part: dydu=cos⁡u\dfrac{dy}{du} = \cos u, dudx=6x\dfrac{du}{dx} = 6x.
  3. Apply the chain rule: dydx=cos⁡u⋅6x=6xcos⁡(3x2+1)\frac{dy}{dx} = \cos u \cdot 6x = 6x\cos(3x^2+1)
Example 2 — Product rule
Differentiate y=x2exy = x^2 e^x.
  1. Let u=x2u = x^2, v=exv = e^x, so u′=2xu' = 2x, v′=exv' = e^x.
  2. Apply the product rule: dydx=u′v+uv′=2xex+x2ex=xex(2+x)\frac{dy}{dx} = u'v + uv' = 2xe^x + x^2e^x = xe^x(2+x)

⚠️ Common mistakes

Forgetting the chain rule — differentiating a composite function without multiplying by the inner derivative, e.g. writing ddx[sin⁡(3x)]=cos⁡(3x)\frac{d}{dx}[\sin(3x)] = \cos(3x) instead of 3cos⁡(3x)3\cos(3x).
Wrong product rule — using (uv)′=u′v′(uv)' = u'v' instead of u′v+uv′u'v + uv'.
Wrong ln⁡x\ln x derivative — ddx[ln⁡x]=1x\frac{d}{dx}[\ln x] = \frac{1}{x}, not 1ln⁡x\frac{1}{\ln x}.

❓ FAQ

What is the difference between implicit and explicit differentiation?
An explicit function can be written as y=f(x)y = f(x) and differentiated directly. An implicit function (e.g. x2+y2=1x^2 + y^2 = 1) cannot be isolated, so you differentiate both sides with respect to xx, applying the chain rule to any yy terms (multiply by dydx\frac{dy}{dx} each time yy appears).
What is a second derivative and what is it for?
The second derivative f′′(x)f''(x) is the derivative of the derivative. It describes concavity: f′′(x)>0f''(x) > 0 means concave up, f′′(x)<0f''(x) < 0 means concave down, and a point where f′′(x)=0f''(x) = 0 and the sign changes is a point of inflection.
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