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AS Maths Edexcel Your notes
7.2 Applications of Differentiation
Contents
7.2.1 Gradients, Tangents & Normals
7.2.2 Increasing & Decreasing Functions
7.2.3 Second Order Derivatives
7.2.4 Stationary Points & Turning Points
7.2.5 Sketching Gradient Functions
7.2.6 Modelling with Differentiation inc. Optimisation
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7.2.1 Gradients, Tangents & Normals
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Gradients, Tangents & Normals
Using the derivative to find the gradient of a curve
To find the gradient of a curve y= f(x) at any point on the curve, substitute the x‑coordinate of the
point into the derivative f'(x)
Using the derivative to find a tangent
At any point on a curve, the tangent is the line that goes through the point and has the same
gradient as the curve at that point
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For the curve y = f(x), you can find the equation of the tangent at the point (a, f(a)) using
y − f ( a ) = f ' ( a ) (x − a )
Using the derivative to find a normal
At any point on a curve, the normal is the line that goes through the point and is perpendicular to
the tangent at that point
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For the curve y = f(x), you can find the equation of the normal at the point (a, f(a)) using
1
y − f ( a) = − (x − a )
f ' ( a)
Exam Tip
The formulae above are not in the exam formulae booklet, but if you understand what
tangents and normals are, then the formulae follow from the equation of a straight line
combined with parallel and perpendicular gradients (see Worked Example below).
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Worked example
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7.2.2 Increasing & Decreasing Functions
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Increasing & Decreasing Functions
What are increasing and decreasing functions?
A function f(x) is increasing on an interval [a, b] if f'(x) ≥ 0 for all values of x such that a< x < b.
If f'(x) > 0 for all x values in the interval then the function is said to be strictly increasing
In most cases, on an increasing interval the graph of a function goes up as x increases
A function f(x) is decreasing on an interval [a, b] if f'(x) ≤ 0 for all values of x such that a < x < b
If f'(x) < 0 for all x values in the interval then the function is said to be strictly decreasing
In most cases, on a decreasing interval the graph of a function goes down as x increases
To identify the intervals on which a function is increasing or decreasing you need to:
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1. Find the derivative f'(x)
2. Solve the inequalities f'(x) ≥ 0 (for increasing intervals) and/or f'(x) ≤ 0 (for decreasing
intervals) Your notes
Exam Tip
On an exam, if you need to show a function is increasing or decreasing you can use either
strict (<, >) or non-strict (≤, ≥) inequalities
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Worked example
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7.2.3 Second Order Derivatives
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Second Order Derivatives
What is the second order derivative of a function?
If you differentiate the derivative of a function (ie differentiate the function a second time) you
get the second order derivative of the function
For a function y = f(x), there are two forms of notation for the second derivative (or second order
derivative)
''
d 2y
f (x ) or 2
dx
Note the positions of the power of 2's in the second version
The second order derivative can also be referred to simply as the second derivative
Similarly, the 'regular' derivative can also be referred to as either the first order derivative or
the first derivative
The second order derivative gives the rate of change of the gradient function (ie of the first
derivative) – this will be important for identifying the nature of stationary points
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Exam Tip
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When finding second derivatives be especially careful with functions that have negative or
fractional powers of x (see Worked Example below).
Mistakes made with fractions or negative signs can build up as you calculate the derivative
more than once.
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Worked example
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7.2.4 Stationary Points & Turning Points
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Stationary Points & Turning Points
What are stationary points?
A stationary point is any point on a curve where the gradient is z ero
To find stationary points of a function f(x)
Step 1: Find the first derivative f'(x)
Step 2: Solve f'(x) = 0 to find the x-coordinates of the stationary points
Step 3: Substitute those x-coordinates into f(x) to find the corresponding y-coordinates
A stationary point may be either a local minimum, a local maximum, or a point of inflection
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Stationary points on quadratics
The graph of a quadratic function (ie a parabola) only has a single stationary point
For an 'up' parabola this is the minimum; for a 'down' parabola it is the maximum (no need to talk
about 'local' here)
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The y value of the stationary point is thus the minimum or maximum value of the quadratic function
For quadratics especially minimum and maximum points are often referred to as turning points
How do I determine the nature of stationary points on other curves?
For a function f(x) there are two ways to determine the nature of its stationary points
Method A: Compare the signs of the first derivative (positive or negative) a little bit to either
side of the stationary point
(After completing Steps 1 - 3 above to find the stationary points)
Step 4: For each stationary point find the values of the first derivative a little bit 'to the left' (ie
slightly smaller x value) and a little bit 'to the right' (slightly larger x value) of the stationary point
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Step 5: Compare the signs (positive or negative) of the derivatives on the left and right of the
stationary point
If the derivatives are negative on the left and positive on the right, the point is a local
minimum
If the derivatives are positive on the left and negative on the right, the point is a local
maximum
If the signs of the derivatives are the same on both sides (both positive or both negative) then
the point is a point of inflection
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Method B: Look at the sign of the second derivative (positive or negative) at the stationary
point
(After completing Steps 1 - 3 above to find the stationary points)
Step 4: Find the second derivative f''(x)
Step 5: For each stationary point find the value of f''(x) at the stationary point (ie substitute the
x-coordinate of the stationary point into f''(x) )
If f''(x) is positive then the point is a local minimum
If f''(x) is negative then the point is a local maximum
If f''(x) is zero then the point could be a local minimum, a local maximum OR a point of
inflection (use Method A to determine which)
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Exam Tip
Usually using the second derivative (Method B above) is a much quicker way of determining
the nature of a stationary point.
However, if the second derivative is z ero it tells you nothing about the point.
In that case you will have to use Method A (which always works – see the Worked
Example).
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Worked example
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7.2.5 Sketching Gradient Functions
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Sketching Gradient Functions
How can I sketch a function's gradient function?
Using your knowledge of gradients and derivatives you can use the graph of a function to sketch
the corresponding gradient function
The behaviour of a function tells you about the behaviour of its gradient function
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Exam Tip
If f(x) is a smooth curve then f'(x) will also be a smooth curve.
Take what you know about f'(x) (based on the table above) and then 'fill in the blanks' in
between.
If all you have is the graph of f(x) you will not be able to specify the coordinates of the y-
intercept or any stationary points of f'(x).
Be careful – points where f(x) cuts the x-axis don't tell you anything about the graph of f'(x)!
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Worked example
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7.2.6 Modelling with Differentiation inc. Optimisation
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Modelling with Differentiation inc. Optimisation
How can I use differentiation to solve modelling questions?
Derivatives can be calculated for any variables – not just y and x
In every case the derivative is a formula giving the rate of change of one variable with respect to
the other variable
Differentiation can be used to find maximum and minimum points of a function (see Stationary
Points)
Therefore it can be used to solve maximisation and minimisation problems in modelling
questions
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Exam Tip
Exam questions on this topic will often be divided into two parts:
First a 'Show that...' part where you derive a given formula from the information in the question
And then a 'Find...' part where you use differentiation to answer a question about the formula
Even if you can't answer the first part you can still use the formula to answer the second part.
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Worked example
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