Ch 7 · 1.7 Differentiation · A Level Mathematics 9709
Topic 7 of 8 in Pure Mathematics 1 — A Level Mathematics 9709 — 78 lessons.
Introduction
A speedometer shows how fast your distance is changing at this instant, not your average for the trip. Differentiation does the same for any curve: it gives the gradient at a point. With it you can find tangents, maxima and minima (the largest area for a fixed fence) and how fast one quantity changes when another does. (Syllabus section 1.7, Paper 1.)
Key ideas — The gradient as a limit
The gradient of the chord joining two points on a curve gets closer to the gradient of the tangent as the points get closer. On y = x^3, the chord from x = 2 to x = 2 + h has gradient
🧑🏫 Teacher's explanation
Imagine you are driving a car along a winding mountain road. If you look at your speedometer, it tells you your speed at that exact instant. However, speed is usually calculated as distance divided by time over an interval. To find your speed at one specific moment, you would need to look at your distance traveled over a smaller and smaller time interval. As that interval shrinks toward zero, the average speed becomes your instantaneous speed.
📖 Definitions
🔍 Worked example — Easy
Find the gradient of the curve y = x^2 at the point where x = 3 using the concept of the limit of a chord.
🔍 Worked example — Standard
Given y = 4x^3 - 2x, find dy/dx and d^2y/dx^2.
🔍 Worked example — Exam-style
A curve has the equation y = x^3 - 3x. Find the coordinates of the stationary points.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
(a) Find dy/dx. (b) Find the coordinates of the stationary points. (c) Determine the nature of each stationary point using the second derivative. [5 marks]
Key ideas — Rules of differentiation
d/dxbig(x^nbig) = nx^n-1 for any rational n
🧑🏫 Teacher's explanation
Imagine you are driving a car. Your speedometer doesn't tell you your average speed for the whole trip; it tells you your 'instantaneous' speed—how fast you are going at that exact split second. Differentiation is the mathematical tool used to find that exact speed (the gradient) at any single point on a curve.
📖 Definitions
🔍 Worked example — Easy
Differentiate y = 5x^3 - 2/x + 7 with respect to x.
🔍 Worked example — Standard
Find dy/dx for y = √(4x^2 + 3).
🔍 Worked example — Exam-style
A curve has the equation y = x^3 - 3x + 2. Find the coordinates of the stationary points and determine their nature using the second derivative.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Tangents and normals
At x = a: the tangent has gradient m = f'(a); the normal is perpendicular to it, gradient -1/m. Use y - y_1 = m(x - x_1) with the point (a, f(a)).
🧑🏫 Teacher's explanation
Imagine you are driving a car along a winding, curvy mountain road. If you were to freeze time at one exact moment, the direction your headlights are pointing is the tangent. It is the straight line that just 'grazes' the curve at that specific point, showing the direction you are heading at that instant. The normal is simply a line that is perfectly perpendicular (at a 90^° angle) to your headlights.
📖 Definitions
🔍 Worked example — Easy
Find the equation of the tangent to the curve y = x^2 + 3x at the point where x = 1.
🔍 Worked example — Standard
Find the equation of the normal to the curve y = √(x) at the point (4, 2).
🔍 Worked example — Exam-style
The curve has equation y = 2x^3 - 5x. Find the gradient of the normal to the curve at the point where x = 1.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Increasing and decreasing functions
f is increasing where f'(x) > 0 and decreasing where f'(x) < 0. To show a function is increasing for all x, show f'(x) > 0 for all x, e.g. by completing the square on f'(x).
🧑🏫 Teacher's explanation
Imagine you are hiking on a mountain trail. If you are walking uphill, your elevation is increasing. In calculus, the derivative f'(x) tells you the exact slope of the ground at any point. If that slope is positive (f'(x) > 0), you are walking upwards (the function is increasing). If the slope is negative (f'(x) < 0), you are walking downwards (the function is decreasing).
📖 Definitions
🔍 Worked example — Easy
Determine the intervals where the function f(x) = x^2 - 4x + 5 is increasing and decreasing.
🔍 Worked example — Standard
Show that f(x) = x^3 + x + 5 is increasing for all real values of x.
🔍 Worked example — Exam-style
A function is defined by f(x) = rac13x^3 - 2x^2 + 3x + 10. Find the values of x for which the function is decreasing.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Stationary points
At a stationary point dy/dx = 0. To find its nature:
🧑🏫 Teacher's explanation
Imagine you are hiking on a mountain range. As you walk, the ground's steepness changes. Sometimes you are climbing up (positive gradient), and sometimes you are walking down (negative gradient). A stationary point is that magical, split-second moment when you are standing on a peak or at the bottom of a valley where the ground is perfectly flat for a tiny instant. At this exact spot, your vertical speed is zero, which in calculus means dy/dx = 0.
📖 Definitions
🔍 Worked example — Easy
Find the coordinates of the stationary point of the curve y = x^2 - 4x + 5 and determine its nature.
🔍 Worked example — Standard
A curve has the equation y = 2x^3 - 3x^2 - 12x + 5. Find the coordinates of its stationary points and state their nature.
🔍 Worked example — Exam-style
A rectangle has a perimeter of 40 cm. Express the area A in terms of its width x, and find the value of x for which the area is a maximum.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Rates of change and connected rates
dy/dx is the rate of change of y with respect to x. When quantities change with time, link them with the chain rule:
🧑🏫 Teacher's explanation
Imagine you are blowing up a balloon. As you blow air into it, the radius r increases, and because the radius is getting bigger, the surface area A also increases. However, the area doesn't increase at the same speed as the radius; it speeds up as the balloon gets larger. This is because the area depends on the radius, and the radius depends on time.
📖 Definitions
🔍 Worked example — Easy
A square has a side length s. Find the rate of change of the area A with respect to time t, in terms of s and ds/dt.
🔍 Worked example — Standard
The volume of a sphere is V = 4/3π r^3. A balloon is being inflated such that the radius r is increasing at a constant rate of 0.2 cm s^-1. Find the rate of increase of the volume when r = 5 cm.
🔍 Worked example — Exam-style
A circle's area is A = π r^2. The area is increasing at a rate of 10 cm^2 s^-1. Find the rate of change of the radius when r = 4 cm.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Worked examples — Example 1: tangent, normal and stationary points
y = x^3 - 3x^2 + 4. (a) Find the equation of the tangent at x = 1. (b) Find the stationary points and their nature.
🔍 Worked example — Easy
Find the equation of the tangent to the curve y = x^2 + 3x at the point where x = 1.
🔍 Worked example — Standard
A curve has the equation y = 2x^3 - 9x^2 + 12x. Find the coordinates of the stationary points and determine their nature.
🔍 Worked example — Exam-style
Find the equation of the normal to the curve y = √(x) at the point where x = 4.
⚠️ Common errors
Worked examples — Example 2: connected rates of change
The radius of a circular ripple increases at 0.2 cm s⁻¹. Find the rate of increase of its area when the radius is 5 cm.
🔍 Worked example — Easy
The volume of a cube is V = s^3, where s is the side length. If the side length is increasing at a rate of 3 cm s^-1, find the rate of increase of the volume when s = 4 cm.
🔍 Worked example — Standard
The radius of a circle increases at 0.2 cm s^-1. Find the rate of increase of its area when the radius is 5 cm.
🔍 Worked example — Exam-style
A spherical balloon is being inflated. The volume is V = 4/3π r^3. The volume is increasing at a constant rate of 100 cm^3 s^-1. Find the rate of increase of the radius when the radius is 5 cm.
⚠️ Common errors
Worked examples — Example 3: a maximum problem
A rectangular pen is built against a wall using 60 m of fencing for the other three sides. Find the greatest area.
🔍 Worked example — Easy
Find the stationary points of the curve y = x^2 - 4x + 5.
🔍 Worked example — Standard
A rectangular area is defined by A = 10x - x^2. Find the value of x that maximizes A and determine its nature using the second derivative.
🔍 Worked example — Exam-style
A rectangular pen is built against a wall using 60 m of fencing for the other three sides. Find the greatest area.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) 12x^2 + 2/x^2 + 3/√(x). (2) 15(3x-1)^4; -32/(2x+1)^3. (3) gradient of tangent = 2, normal gradient -tfrac12: x + 2y + 1 = 0. (4) 6x^2 + 6x - 12 = 6(x+2)(x-1) = 0: (-2, 20) maximum, (1, -7) minimum. (5) f'(x) = 3x^2 + 6x + 6 = 3(x+1)^2 + 3 > 0 for all x. (6) dV/dr = 4π r^2 = 400π; dr/dt = 50/400π = 1/8π ≈ 0.0398 cm s⁻¹.