1. What is the differentiation.
Essential Mathematics for Economics and Business, 4th Edition
What is the differentiation.
© John Wiley and Sons 2013
© John Wiley and Sons 2013

2. Slopes/rates of change Recall linear functions
For linear functions slope is the change in y per unit increase in x
Slope is the rate of change
The slope or rate of change is the same over any interval of any size
See the following example

3. Example: calculate the rate of change for a linear function
Suppose the distance, (x m) that a ball after y (seconds) is given by the equation y = x
Slope is the rate of change
The average speed of the ball is 1 meter per second, between the 1st and 4th sec
Δy = 4 -1 = 3
y (distance in m)
Δx = 4-1=3
x (time in seconds)

4. Example: the rate of change for a linear function.. continued
Now take a different interval, 1 to 3 sec
Slope is the rate of change =
The slope of this linear function (speed) is 1 m/s when measured over any interval of any length!
Average speed is 1 meter per second
y (distance in m)
Δy = 3-1=2
Δx =3-1=2
x (time in seconds)

5. The slope of a curve varies along the curve
Suppose the distance y that a ball travels after x seconds is given by the equation y = x2
The slope varies along the curve.
y (distance in m)
x (time in seconds)

6. How do find the rate of change (speed) at x = 1?
Start by getting the average slope over progressively smaller intervals where one end is fixed at x = 1.
The average slope over the smallest possible interval at x = 1approximates the slope at x = 1.
y (distance in m)
x (time in seconds)

7. What is the rate of change (speed) at x = 1
Start by getting the average slope over progressively smaller intervals, where one end is fixed at x = 1.
The average slope over the smallest possible interval at x = 1 approximates the slope at x = 1.
y (distance in m)
x (time in seconds)

8. Average slope of the curve between x = 1 and x = 3
or the average speed between 1st and 3rd second is 4 m/sec
Slope of chord BC C
Δy = 9-1=8
y (distance in m) B
Δx = 3-1=2
x (time in seconds)

9. Average slope of the curve between x = 1 and x = 2.5
or the average speed between 1 and 2.5 seconds is 3.5 m/sec
Slope of chord BC C
y (distance in m)
Δy = =5.25 B
Δx = 2.5-1=1.5
x (time in seconds)

10. Average slope of the curve between x = 1 and x = 2
or the average speed between the 1st and 2nd seconds is 3 m/sec
Slope of chord BC C
y (distance in m)
Δy = 4-1=3 B
Δx = 2 - 1=1
x (time in seconds)

11. Average slope of the curve between x = 1 and x = 1.5
or the average speed between 1 and 1.5 seconds is 2.5 m/sec
Slope of chord BC C B
Δy = =1.25
Δx = = 0.5

12. Average slope of the curve between x = 1 and x = 1.1
or the average speed between 1 and 1.1 seconds is 2.1 m/sec
Slope of chord BC C B
Δy = =0.21
Δx = = 0.1

13. The slope of a curve at x = 1
Zero divided zero is not defined!
So….
we need another method to find the slope at a point
Slope of chord at B B

14. Summary, so far y = x2 is the distance travelled, y after x seconds
Time interval: between
Average speed
1st and 3rd second
4 m/s
1 and 2.5 seconds
3.5 m/s
1st and 2nd second
3 m/s
1 and 1. 5 second
2.5 m/s
1 and 1.1 second
2.1 m/s
at 1 second
Not defined ???
Slope (average speed) is approaching the value 2 as the interval Δx is becoming progressively smaller towards x = 1

15. Find slope of a curve at a point, graphically
Slope of the curve at B = slope of the tangent at B
But…the accuracy of graphical methods is limited!
Measure the slope (m) of the tangent line B
h = 4.0
d = 2.0

16. The slope of a curve at a point is determined analytically by differentiation.
Method: given the equation of the curve, y = x2 First, find the derivative of y = x2 (see Worked Example 6.1 for proof ) Then evaluate the derivative at the given point At x = 1,

17. The slope of a curve at a point, graphically and analytically
curve y = x2
At x = 1, the slope,
m =
Slope of the curve at B = slope of the tangent at B
Graphically, measure the slope (m) of the tangent line B
h = 4.0
d = 2.0

18. Summary: Slope of chord = 4, the average slope of the curve between
x = 1 and x = 3
Slope of chord =3.5, the average slope of the curve between
x = 1 and x = 2.5
Slope of chord =3, the average slope of the curve between
x = 1 and x = 2
Slope of chord =2.5, the average slope of the curve between
x = 1 and x = 1.5
Slope of curve at
x = 1 is the same as the slope of the tangent at x =1
By differentiation
slope of curve at
x = 1 is

19. Procedure
The slope of a curve at a point is determined analytically by differentiation
Method consists of two simple steps.
Given the curve, y = f(x), find its derivative ,
Evaluate the derivative at the given point
First
Next
The slope of a curve at a point is also described as the instantaneous rate of change

20. At x = -1, dy/dx = 2(-1) = - 2 At x = 1.5, y = 2.25, dy/dx = 2(1.5) =3
Worked Example 6.1
Figure 6.3:
Equation of the curve y = x2
Equation for slope of curve, dy/dx = 2x
At x = -1, dy/dx = 2(-1) = - 2
At x = 1.5, y = 2.25, dy/dx = 2(1.5) =3 -2
-1.5 -1
-0.5
0.5 1
1.5 2
2.5 3
3.5 4 -3
-2.5 x y
(i) y = 3x -2.25
(ii) y = -2x - 1
Figure 6.3: Slope of curve y = x2 at x = -1 and x = 1.5

21. Equation of the curve y = x2 Equation for slope of curve, dy/dx = 2x
Worked Example 6.1
Figure 6.3:
Equation of the curve y = x2
Equation for slope of curve, dy/dx = 2x
At x = -1, dy/dx = 2(-1) = - 2
At x = 1.5, y = 2.25, dy/dx = 2(1.5) 1 2 3 4 5 6 7 8 9 -3 -2 -1
Figure 6.3: Slope of curve y = x2 at x = -1 and x = 1.5