Ch 1 · 14.1–14.4 Differentiation · O Level Additional Mathematics 4037
Topic 1 of 4 in Calculus — O Level Additional Mathematics 4037 — 54 lessons.
Introduction
How fast is a population growing right now? How steep is a road at this exact point? Differentiation measures an instantaneous rate of change: the gradient of a curve at a point. This lesson gives the rules; the next lessons use them for tangents, maxima, rates and motion. No formulas for calculus are given in the exam, so these must be learned. (Syllabus 14.1–14.4.)
Key ideas — The derived function
The gradient of the chord between (x, y) and (x + delta x, y + delta y) is delta y/delta x. As delta x → 0 the chord becomes the tangent, and its gradient becomes the derivative:
🧑🏫 Teacher's explanation
Imagine you are driving a car. If you look at your average speed over an hour, you are calculating the gradient of a chord between your starting point and your ending point. However, if you look at your speedometer at one exact, frozen moment in time, you are looking at your instantaneous speed. In calculus, the 'derivative' is that speedometer reading.
📖 Definitions
🔍 Worked example — Easy
Find dy/dx if y = 5x^3 - 2x + 7.
🔍 Worked example — Standard
Differentiate y = sin(4x) + e^2x.
🔍 Worked example — Exam-style
Find dy/dx for y = x^2 + 1/x - 1.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Standard derivatives
Angles are in radians. Constant multiples and sums are differentiated term by term; rewrite roots and fractions as powers first (√(x) = x^1/2, 3/x^2 = 3x^-2).
🧑🏫 Teacher's explanation
Imagine you are driving a car. Your position changes over time, but your speedometer tells you how fast you are moving at a single, exact moment. Differentiation is the mathematical tool that calculates that 'instantaneous' rate of change. Instead of looking at how much you traveled over an hour, we look at how much you change over a tiny, tiny interval delta x → 0.
📖 Definitions
🔍 Worked example — Easy
Differentiate y = 4x^3 - 2/x + 5 with respect to x.
🔍 Worked example — Standard
Find dy/dx for y = e^3x sin(x).
🔍 Worked example — Exam-style
Differentiate y = ln x/x^2 with respect to x.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — The chain rule
For a function of a function, dy/dx = dy/du × du/dx: differentiate the outside, keep the inside, multiply by the derivative of the inside.
🧑🏫 Teacher's explanation
Imagine you are watching a movie. If the actor moves twice as fast as the camera is panning, and the camera is moving three times as fast as a person standing on the sidewalk, how fast is the actor moving relative to the sidewalk? You multiply the rates: 2 × 3 = 6. This is the essence of the Chain Rule. It allows us to find the rate of change of a 'composite' function—a function tucked inside another function.
📖 Definitions
🔍 Worked example — Standard
Find dy/dx for y = ln(5x^2 + 3)
🔍 Worked example — Exam-style
Differentiate y = cos(e^2x)
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — The product rule
d/dx(uv) = udv/dx + vdu/dx
🧑🏫 Teacher's explanation
Imagine you are calculating the area of a growing rectangle. The area is A = L × W. If both the length (L) and the width (W) are changing at the same time, the rate at which the area changes depends on how fast the length is growing (while the width stays constant for a split second) plus how fast the width is growing (while the length stays constant for a split second).
📖 Definitions
🔍 Worked example — Standard
Find dy/dx if y = e^x sin x.
🔍 Worked example — Exam-style
Differentiate f(x) = x^2 ln(x) with respect to x.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — The quotient rule
d/dx (u/v ) = vdu/dx - udv/dx/v^2
🧑🏫 Teacher's explanation
Imagine you are tracking the speed of a changing ratio, like the density of a liquid as you pour it into a container. If both the amount of liquid and the volume of the container are changing at the same time, you can't just differentiate them separately. You need a rule that accounts for how the 'top' part changes relative to the 'bottom' part.
📖 Definitions
🔍 Worked example — Standard
Find dy/dx for y = sin x/x
🔍 Worked example — Exam-style
Differentiate f(x) = frace^3xx^2 + 4
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples
y = 4x^3 - 6/x + 2√(x): dy/dx = 12x^2 + 6/x^2 + 1/√(x).
🔍 Worked example — Standard
Find dy/dx for y = √(x) ln x.
🔍 Worked example — Exam-style
Find the second derivative d^2y/dx^2 for y = e^2x + sin x.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) 20x^3 + 6x^-3. (2) 4x/√(4x^2 + 1). (3) 3x^2e^-2x - 2x^3e^-2x = x^2e^-2x(3 - 2x). (4) xcos x - sin x/x^2. (5) -sin x/cos x = -tan x.