Ch 4 · 2.4 / 3.4 Differentiation · A Level Mathematics 9709
Topic 4 of 9 in Pure Mathematics 2 and 3 — A Level Mathematics 9709 — 69 lessons.
Introduction
Paper 1 differentiated powers of x. Real models also use exponentials (growth and decay), logarithms and trigonometric functions (waves), multiplied and divided together. This lesson adds those derivatives, the product and quotient rules, and two new ways a curve can be given: parametrically (as the path of a moving point) and implicitly (like a circle's equation). Sections 2.4 (Paper 2) and 3.4 (Paper 3) are the same except that only Paper 3 includes tan^-1x.
Key ideas — Standard derivatives (P2, P3)
x must be in radians for the trigonometric results. With the chain rule: d/dxe^f(x) = f'(x)e^f(x), d/dxln f(x) = f'(x)/f(x), and d/dxsin(ax + b) = acos(ax + b). Simplify logs before differentiating: lnx^2/x+1 = 2ln x - ln(x + 1). (Derivatives of sin^-1x and cos^-1x are not required.)
🧑🏫 Teacher's explanation
Differentiation is the mathematical way of finding the 'instantaneous rate of change'. If you imagine a car driving along a curvy road, the derivative tells you exactly how fast the car's position is changing at one specific moment, rather than over a long interval. Think of it like a high-speed camera capturing a single frame; the derivative is the 'slope' of the path at that exact micro-second.\n\nWhen functions get complicated, we use 'rules' to break them down. The Product Rule handles two functions multiplied together, the Quotient Rule handles division, and the Chain Rule handles 'nested' functions (a function inside another). For trigonometric functions, remember that calculus only works when your angles are in radians; if you try to differentiate sin(x) where x is in degrees, the math breaks!\n\nAlways look to simplify before you differentiate. If you see ln(x/y), don't jump straight to the quotient rule; use log laws to turn it into ln x - ln y first. It makes the calculus much smoother.
📖 Definitions
🔍 Worked example — Easy
Differentiate y = 5e^3x + sin(2x).
🔍 Worked example — Standard
Find dy/dx for the curve defined by x^2 + y^2 = 10.
🔍 Worked example — Exam-style
A curve is defined parametrically by x = t^2 and y = tan^-1(t). Find dy/dx in terms of t.
⚠️ Common errors
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🎯 Exam question
Key ideas — Product rule (P2, P3)
d/dx(uv) = udv/dx + vdu/dx
🧑🏫 Teacher's explanation
Imagine you are calculating the area of a growing rectangle where both the length and the width are changing at the same time. If you want to know how fast the total area is increasing, you can't just look at one side; you have to account for how much the length adds to the area while the width stays the same, plus how much the width adds while the length stays the same.
📖 Definitions
🔍 Worked example — Standard
Find dy/dx for y = x^2 e^3x.
🔍 Worked example — Exam-style
A function is defined as f(x) = ln(x) · cos(x). Find the gradient of the curve at x = π/2.
⚠️ Common errors
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🎯 Exam question
Key ideas — Quotient rule (P2, P3)
d/dx (u/v ) = vdu/dx - udv/dx/v^2
🧑🏫 Teacher's explanation
When you have two functions multiplied together, you use the Product Rule. But when one function is being divided by another, it's like a tug-of-war where the denominator is trying to pull the whole value down while the numerator tries to push it up. The Quotient Rule is simply a specialized version of the Product Rule that handles this 'division' relationship directly.
📖 Definitions
🔍 Worked example — Exam-style
A function is defined as f(x) = frace^2xsin x. Find the exact value of f'(π/4).
⚠️ Common errors
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🎯 Exam question
Key ideas — Parametric differentiation (P2, P3)
If x and y are both given in terms of a parameter t:
🧑🏫 Teacher's explanation
Imagine you are watching a race car on a video screen. The camera shows the car's position on the x-axis and the y-axis separately over time t. If you want to know the car's direct path (the slope dy/dx), you can't just look at x or y alone; you have to see how fast y is changing relative to t and how fast x is changing relative to t.
📖 Definitions
🔍 Worked example — Easy
Find dy/dx for the parametric equations x = t^2 and y = 3t + 5.
🔍 Worked example — Standard
A curve is defined by x = e^t and y = sin(t). Find the gradient of the tangent to the curve at t = π/2.
🔍 Worked example — Exam-style
Find the equation of the normal to the curve x = t^3 and y = t^2 at the point where t = 2.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Implicit differentiation (P2, P3)
When x and y are mixed in one equation, differentiate every term with respect to x, treating y as a function of x:
🧑🏫 Teacher's explanation
Imagine you are tracking two moving objects, x and y. In standard differentiation, we assume y depends directly on x. However, in implicit differentiation, x and y are tangled together in an equation like x^2 + y^2 = 25. We cannot easily write y = dots without dealing with ± signs, so we treat y as a 'hidden' function of x.
📖 Definitions
🔍 Worked example — Standard
Find dy/dx for the equation x^2y + y^3 = 10.
🔍 Worked example — Exam-style
Find the gradient of the tangent to the curve x^2 + xy + y^2 = 7 at the point (1, 2).
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Worked examples — Example 1: product with a composite
Differentiate y = xe^1 - x^2 and find the stationary points.
🔍 Worked example — Easy
Differentiate y = x^2 sin(x).
🔍 Worked example — Exam-style
Differentiate y = x e^1-x^2 and find the stationary points.
⚠️ Common errors
Worked examples — Example 2: quotient
y = 2x + 1/x - 3: dy/dx = 2(x - 3) - (2x + 1)(1)/(x - 3)^2 = -7/(x - 3)^2.
⚠️ Common errors
Worked examples — Example 3: parametric tangent
A curve has x = t - e^2t, y = t + e^2t. Find the tangent at t = 0.
🔍 Worked example — Easy
A curve is defined by x = t^2 and y = 3t. Find dy/dx in terms of t.
🔍 Worked example — Standard
A curve has parametric equations x = t - e^2t and y = t + e^2t. Find the equation of the tangent to the curve at t = 0.
🔍 Worked example — Exam-style
A curve is defined by x = sin t and y = cos(2t). Find the gradient of the tangent at t = π/4.
⚠️ Common errors
Worked examples — Example 4: implicit gradient
The curve x^2 + y^2 = xy + 7. Find the gradient at each point where x = 1.
🔍 Worked example — Easy
Find dy/dx for the equation x^2 + y^2 = 10.
🔍 Worked example — Standard
Find the gradient of the curve x^2 + y^2 = xy + 7 at the point where x = 1 and y = 3.
🔍 Worked example — Exam-style
A curve is defined by x^2 + y^2 = xy + 7. Find the gradient of the curve at the point (1, -2).
⚠️ Common errors
Worked examples — Example 5: inverse tangent (P3)
d/dx (xtan^-1x ) = tan^-1x + x/1 + x^2.
🔍 Worked example — Standard
Differentiate y = xtan^-1x.
🔍 Worked example — Exam-style
Find the gradient of the curve y = xtan^-1x at the point where x = 1.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) (a) e^3x(3sin 2x + 2cos 2x); (b) 2x/x^2 + 4; (c) 3sec^2(3x - 1). (2) 1 - ln x/x^2; stationary at x = e: (e,\ tfrac1e ). (3) dy/dx = -tfrac32cot t; at t = tfracπ4: point (sqrt2, 3/sqrt2 ), gradient -tfrac32, normal gradient tfrac23: y - 3sqrt2/2 = tfrac23 (x - sqrt2 ). (4) 8 + 64 = 72 = 9(2)(4) ✓; 3x^2 + 3y^2y' = 9y + 9xy' ⇒ y' = 3y - x^2/y^2 - 3x = 12 - 4/16 - 6 = tfrac45. (5) 2/1 + 4x^2.