O-Level MathOctober 2026 • 12 min read

O-Level A-Math Calculus: Differentiation & Integration Explained

Complete guide to O-Level A-Math calculus. Differentiation rules, integration techniques, and common exam mistakes for Sec 3-4 students taking Syllabus 4049.

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Elite Pro Tip

"Calculus is roughly 30% of the A-Math syllabus and the single largest source of marks in Paper 1. Most students lose marks not on the hard problems, but on the basic rules applied carelessly — chain rule sign errors and forgetting +C in integration are worth more marks than any trick question."

Dr. Chen

Curriculum Architect & PhD in Education

Why Calculus Matters for A-Math

Calculus (Differentiation and Integration) carries approximately 30% of the A-Math syllabus weightage. It appears in both Paper 1 and Paper 2, and the marks are relatively predictable — the same question types appear year after year. Mastering calculus rules and recognizing when to apply each one is the highest-return revision activity for A-Math.

Interactive Bar Model

Differentiation

Integration

Applications

Kinematics

Elite Pro Tip

Calculus topics carry roughly 30% of A-Math marks. Differentiation and integration rules must be automatic — recognition under exam pressure is what separates A1 from B3.

Part 1: Differentiation

Differentiation finds the rate of change of a function. Geometrically, it gives the gradient of the tangent at any point on a curve.

The Core Rules

Rule	Formula	When to Use
Power Rule	d/dx(xⁿ) = nxⁿ⁻¹	Any single power of x
Constant Rule	d/dx(c) = 0	Derivative of a constant
Sum Rule	d/dx(f + g) = f' + g'	Terms added/subtracted
Product Rule	d/dx(fg) = f'g + fg'	Two functions multiplied
Quotient Rule	d/dx(f/g) = (f'g - fg')/g²	Two functions divided
Chain Rule	d/dx(f(g(x))) = f'(g(x)) · g'(x)	Composite functions (function of a function)

The Chain Rule in Practice

The chain rule is the most frequently tested and most frequently missed rule. Any time you see a function inside another function, you need the chain rule.

Example: Differentiate y = (3x² + 1)⁵

▸Identify inner function: u = 3x² + 1
▸Outer function: y = u⁵
▸Differentiate outer: dy/du = 5u⁴
▸Differentiate inner: du/dx = 6x
▸Multiply: dy/dx = 5u⁴ · 6x = 30x(3x² + 1)⁴

Common mistake: Forgetting to multiply by the derivative of the inner function. If you wrote 5(3x² + 1)⁴ instead of 30x(3x² + 1)⁴, you lost the chain rule factor entirely.

Applications of Differentiation

Application	What You Find	Key Formula
Gradient at a point	Slope of tangent	Substitute x into dy/dx
Stationary points	Max, min, or point of inflection	Set dy/dx = 0 and solve
Increasing/Decreasing	Sign of dy/dx	dy/dx > 0 means increasing
Kinematics	Velocity from displacement	v = ds/dt, a = dv/dt

Stationary Points: The Full Method

▸Find dy/dx
▸Set dy/dx = 0 and solve for x
▸Find the corresponding y values
▸Use the second derivative (d²y/dx²) or a sign change table to classify each point

Example: Find and classify the stationary points of y = x³ - 6x² + 9x

▸dy/dx = 3x² - 12x + 9
▸Set to 0: 3(x² - 4x + 3) = 0 → 3(x-1)(x-3) = 0 → x = 1 or x = 3
▸y(1) = 1 - 6 + 9 = 4, y(3) = 27 - 54 + 27 = 0
▸d²y/dx² = 6x - 12. At x=1: -6 (negative → local max). At x=3: +6 (positive → local min)

Part 2: Integration

Integration is the reverse of differentiation. If differentiation gives the gradient, integration gives the original function from its gradient.

The Core Rules

Rule	Formula	When to Use
Power Rule	∫xⁿ dx = xⁿ⁺¹/(n+1) + C	Any power of x (n ≠ -1)
Constant Multiple	∫kf(x) dx = k∫f(x) dx	Constant factor outside
Sum Rule	∫(f + g) dx = ∫f dx + ∫g dx	Terms added/subtracted
Chain Rule (reverse)	∫f'(g(x))·g'(x) dx = f(g(x)) + C	When you can spot the derivative inside

The +C is not optional. The constant of integration represents all possible original functions whose derivative matches the given expression. Omitting +C in an indefinite integral costs marks every time.

Definite Integration

A definite integral ∫ₐᵇ f(x) dx gives the area under the curve between x = a and x = b.

Method:

▸Integrate to get F(x)
▸Evaluate F(b) - F(a)
▸No +C needed (it cancels out)

Example: Find ∫₀² (3x² + 1) dx

▸∫(3x² + 1) dx = x³ + x
▸Evaluate at 2: 8 + 2 = 10
▸Evaluate at 0: 0 + 0 = 0
▸Answer: 10 - 0 = 10

Area Between Curves and the x-axis

When a curve crosses the x-axis within the integration limits, you must split the integral at the crossing point. The area below the x-axis is counted as negative if you integrate directly — to get total geometric area, take the absolute value of each section and add.

Part 3: Kinematics

Kinematics connects calculus to motion. If s = displacement, v = velocity, a = acceleration:

▸v = ds/dt (velocity is the derivative of displacement)
▸a = dv/dt (acceleration is the derivative of velocity)
▸s = ∫v dt (displacement is the integral of velocity)

Typical exam question: "A particle moves such that s = 2t³ - 9t² + 12t. Find (a) the velocity when t = 2, (b) the acceleration when t = 2, (c) the distance travelled in the first 3 seconds."

(a) v = ds/dt = 6t² - 18t + 12. At t=2: v = 24 - 36 + 12 = 0 m/s (b) a = dv/dt = 12t - 18. At t=2: a = 24 - 18 = 6 m/s² (c) Find when v = 0: 6t² - 18t + 12 = 0 → t² - 3t + 2 = 0 → t = 1 or t = 2

▸s(0) = 0, s(1) = 2 - 9 + 12 = 5, s(2) = 16 - 36 + 24 = 4, s(3) = 54 - 81 + 36 = 9
▸Distance = |s(1)-s(0)| + |s(2)-s(1)| + |s(3)-s(2)| = 5 + 1 + 5 = 11 units

Common Exam Mistakes

Mistake	How to Avoid It
Forgetting the chain rule factor	Circle the inner function, differentiate it, multiply
Omitting +C in indefinite integrals	Write +C before you start — make it automatic
Wrong sign in quotient rule	Memorize: (f'g - fg')/g² — the numerator is f'g MINUS fg'
Not splitting area at x-axis crossings	Always sketch the curve first. Check for sign changes.
Confusing distance and displacement in kinematics	Distance = sum of absolute values of each segment. Displacement = s(final) - s(initial).