1
Differentiation and Integration
Differentiation and
Integration together are called Calculus.
Maths revision course by Miriam Hanks

2
Differentiation
How do you know when to differentiate?
Differentiate if the question says:
 Derive
 Find the derivative



Find the rate of change
Find dy dx or f ’(x)
Maximise or minimise
Maths revision course by Miriam Hanks

3
Differentiation
How do you prepare for differentiation?
 Change x to
 x
1
2
Get rid of fractions with x on the bottom: eg
3

3 x

2 x
2
 Multiply out brackets
Maths revision course by Miriam Hanks

4
Differentiation
How do you differentiate?
Bring the power to the front (to multiply)
Subtract 1 from the power
Maths revision course by Miriam Hanks

5
Differentiation in real life
Why was differentiation developed originally?
In Isaac Newton's day, shipwrecks occurred regularly because the ship was not where the captain thought it was using only the stars for guidance.
Calculus was developed to improve navigation techniques.
Maths revision course by Miriam Hanks

6
Differentiation in real life
Differentiation is used in economics and business to find optimum solutions. eg to use the minimum amount of material to make the packaging, or to maximise efficiency of products or to minimise their costs.
Maths revision course by Miriam Hanks

7
Differentiation and Integration
Click for calculus video
Maths revision course by Miriam Hanks

8
Graphs of the derivative
If you are asked to draw the graph of the derivative, keep in mind that this is a graph of the gradient.
 Start by marking points on the axes where there are turning points on the original graph.
 Now look at the gradient before and after each of these points: Is it up (in which case, draw above the x- axis) or down (draw below x-axis)?
 Make sure you curve is smooth
Maths revision course by Miriam Hanks

9
Integration
How do you know when to integrate?
Integrate if the question says:
 Find the integral
 Find the area under a curve



Find the area between 2 curves
Find
 dx
Find y when you are given dy dx
Maths revision course by Miriam Hanks

10
Integration
How do you prepare for integration?
 Change x to
 x
1
2
Get rid of fractions with x on the bottom: eg
3

3 x

2 x
2
 Multiply out brackets
Maths revision course by Miriam Hanks

11
Integration
How do you integrate?
Add 1 to the power
Divide by the new power
+ c
Maths revision course by Miriam Hanks

12
Integration in real life
Integration is used to analyse crash test data, to improve safety of cars.
It is also used by engineers to find the force of water from the surface area of a reservoir, to design dams.
Maths revision course by Miriam Hanks

13
Further Calculus
To differentiate or integrate a trig function, use the formula sheet: sin ax cos ax differentiates to a cos ax
-a sin ax sin ax cos ax integrates to
1
a cos ax
1 sin ax a
Maths revision course by Miriam Hanks

14
Further Calculus
To differentiate or integrate a bracket with a power:
1.
Think of the whole bracket as an “x” and differentiate or integrate it as usual
2.
Multiply by the derivative of the bracket
(if differentiating) or Divide by the derivative of the bracket
(if integrating)
Maths revision course by Miriam Hanks

15
Integration
Click for an integration joke
Maths revision course by Miriam Hanks