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CIE IGCSE Maths: Extended Your notes
2.18 Differentiation
Contents
2.18.1 Differentiation
2.18.2 Applications of Differentiation
2.18.3 Problem Solving with Differentiation
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2.18.1 Differentiation
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Differentiation
What is differentiation?
Differentiation is part of the branch of mathematics called Calculus
It is concerned with the rate at which changes takes place – so has lots of real‑world uses:
The rate at which a car is moving (its speed)
The rate at which a virus spreads amongst a population
To begin to understand differentiation you’ll need to understand gradients
How are gradients related to rates of change?
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Gradient generally means steepness.
For example, the gradient of a road up the side of a hill is important to lorry drivers
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On a graph the gradient refers to how steep a line or a curve is
It is really a way of measuring how fast y changes as x changes
This may be referred to as the rate at which y
So gradient describes the rate at which change happens
How do I find the gradient of a curve using its graph?
For a straight line the gradient is always the same (constant)
Recall y = mx + c, where m is the gradient
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Your notes
For a curve the gradient changes as the value of x changes
At any point on the curve, the gradient of the curve is equal to the gradient of the tangent at that point
A tangent is a straight line that touches the curve at one point
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Your notes
How do I find the gradient of a curve using algebra?
This is really where the fun begins!
Drawing tangents each time you want the gradient of a curve is too much effort
It would be great if you could do it using algebra instead
The equation of a curve can be given in the form y = f (x )
Inputting x-coordinates gives outputs of y-coordinates
It is possible to create an algebraic function that take inputs of x-coordinates and gives outputs of
gradients
All of this is done without needing to sketch any graphs
This type of function has a few commonly used names:
The gradient function
The derivative
The derived function
dy
The way to write this function is
dx
This is pronounced "dy by dx"
In function notation, it can be written f'(x )
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pronounced f-dashed-of-x
dy
To get from y = f (x ) to = f'(x ) you need to do an operation called differentiation Your notes
dx
Differentiation turns curve equations into gradient functions
The main rule for differentiation is shown
This looks worse than it is!
For powers of x
STEP 1 Multiply the number in front by the power
STEP 2 Take one off the power (reduce the power by 1)
2x6 differentiates to 12x5
Note the following:
kx differentiates to k
so 10x differentiates to 10
any number on its own differentiates to zero
so 8 differentiates to 0
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Your notes
How do I use the gradient function to find gradients of curves?
Find the x-coordinate of the point on the curve you're interested in
dy
Use differentiation to find the gradient (derived) function,
dx
Substitute the x-coordinate into the gradient (derived) function to find the gradient
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Examiner Tip
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When differentiating long awkward expressions, write each step out fully and simplify the numbers
after
Don't forget to write the left-hand sides of y = .... and = ... to avoid mixing up the curve
equation with the gradient function
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Worked example
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2.18.2 Applications of Differentiation
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Finding Stationary Points & Turning Points
What is a turning point?
The easiest way to think of a turning point is that it is a point at which a curve changes from moving
upwards to moving downwards, or vice versa
Turning points are also called stationary points
stationary means the gradient is zero (flat) at these points
At a turning point the gradient of the curve is zero.
If a tangent is drawn at a turning point it will be a horizontal line
Horizontal lines have a gradient of zero
This means substituting the x-coordinate of a turning point into the gradient function (aka derived
function or derivative) will give an output of zero
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Your notes
How do I find the coordinates of a turning point?
STEP 1: Solve the equation of the gradient function (derivative / derived function) equal to zero
dy = 0
ie. solve
dx
This will find the x-coordinate of the turning point
STEP 2: To find the y-coordinate of the turning point, substitute the x-coordinate into the equation of
the graph, y = ...
not into the gradient function
Examiner Tip
Remember to read the questions carefully (sometimes only the x-coordinate of a turning point is
required)
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Worked example
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Classifying Stationary Points
What are the different types of stationary points? Your notes
You can see from the shape of a curve the different types of stationary points
You need to know two different types of stationary points (turning points):
Maximum points (this is where the graph reaches a “peak”)
Minimum points (this is where the graph reaches a “trough”)
These are sometimes called local maximum/minimum points as other parts of the graph may still reach
higher/lower values
How do I use graphs to classify which is a maximum point and which is a minimum point?
You can see and justify which is a maximum point and which is a minimum point from the shape of a
curve...
... either from a sketch given in the question
... or a sketch drawn by yourself
(You may even be asked to do this as part of a question)
... or from the equation of the curve
For parabolas (quadratics) it should be obvious ...
... a positive parabola (positive x2 term) has a minimum point
... a negative parabola (negative x2 term) has a maximum point
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Your notes
Cubic graphs are also easily recognisable ...
... a positive cubic has a maximum point on the left, minimum on the right
... a negative cubic has a minimum on the left, maximum on the right
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Your notes
How do I use the second derivative to classify which is a maximum point and which is a
minimum point?
d2y
The second derivative, , is the derivative-of-the-derivative
dx 2
dy d2y
differentiate the expression for to get the expression for
dx dx 2
this is the same as differentiating the original equation for y twice
A quick algebraic test to find out the turning point (that does not require sketching) is as follows
d2y
If the stationary point is at x = a , substitute x = a into the expression for to get a
dx 2
numerical value...
d2y
...if this value is negative, < 0 , the stationary point is a maximum point
dx 2
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d2y
...if this value is positive, > 0 , the stationary point is a minimum point
dx 2 Your notes
d2y
If the value is zero, = 0 , then unfortunately the test has failed
dx 2
a zero means it could be any out of a max, min, or other types (stationary points-of-
inflection)
go back to sketching the graph to classify the stationary point(s)
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Worked example
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2.18.3 Problem Solving with Differentiation
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Problem Solving with Differentiation
What problems could involve differentiation?
Differentiation allows analysis of how one quantity changes as another does
The derived function (gradient function / derivative) gives a measure of the rate of change
Problems involving a variable quantity can involve differentiation
How the area of a rectangle changes as its length varies
How the volume of a cylinder changes as its radius varies
How the position of a car changes over time (i.e. its speed)
Problems based on the graph of a curve may also arise
The distance between two turning points
The area of a shape formed by points on the curve such as turning points and axes intercepts
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Your notes
How do I solve problems involving differentiation?
Problems generally fall into two categories:
1. Graph-based problems
These problems are based around the graph of a curve and its turning points
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2. Maximum/Minimum problems
The maximum or minimum values have a meaning in the question
e.g. the maximum volume of a box made from a flat sheet of material
e.g. the minimum height of water in a reservoir
These are sometimes called optimisation problems
The maximum or minimum value gives the optimal (ideal/best) solution to the problem
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Examiner Tip
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Diagrams can help – if you are not given one, sketch one and add to it as you go along
Make sure you know how to find the areas and volumes of basic shapes, eg. area of squares,
rectangles, triangles, circles, volume of cubes, cuboids and cylinders.
Early parts of questions often ask you to “show that” a result is true – even if you can’t do this part
of the question, you can use the answer shown to continue with the rest of the question
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Worked example
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