1. Differential calculus

2. Differential calculus is concerned with the rate at which things change.
For example, the speed of a car is the rate at which the distance it travels changes with time.
First we shall review the gradient of a straight line graph, which represents a rate of change.

3. Gradient of a straight line graph
The gradient of the line between two points (x1, y1) and (x2, y2) is
where m is a fixed number called a constant.
A gradient can be thought of as the rate of change of y with respect to x.

4. Gradient of a curve
A curve does not have a constant gradient. Its direction is continuously changing, so its gradient will continuously change too.
y = f(x)
The gradient of a curve at any point on the curve is defined as being the gradient of the tangent to the curve at this point.

5. We cannot calculate the gradient of a tangent directly
A tangent is a straight line, which touches, but does not cut, the curve. x y O A
Tangent to
the curve
at A.
, as we know only one point on the tangent and we require two points to calculate the gradient of a line.
We cannot calculate the gradient of a tangent directly

6. Using geometry to approximate to a gradient Look at this curve.B1 B2
Tangent to
the curve
at A. B3 x O
Look at the chords AB1, AB2, AB3, . . .
For points B1, B2, B3, that are closer and closer to A the sequence of chords AB1, AB2, AB3, move closer to becoming the tangent at A.
The gradients of the chords AB1, AB2, AB3, move closer to becoming the gradient of the tangent at A.

7. A numerical approach to rates of change
Here is how the idea can be applied to a real example. Look at the section of the graph of y = x2 for 2 > x > 3. We want to find the gradient of the curve at A(2, 4).
Chord x
changes
from y
gradient
AB1
AB2
AB3
AB4
AB5
B1 (3, 9)
2 to 3
4 to 9
= 5
B2 (2.5, 6.25)
2 to 2.5
4 to 6.25
= 4.5
B3 (2.1, 4.41)
B4 (2.001, )
2 to 2.1
4 to 4.41
= 4.1
A (2, 4)
Complete the table
The gradient of the chord AB1 is
2 to 2.001
4 to
= 4.001
2 to
4 to

8. As the points B1, B2, B3, . . . get closer and closer to A the gradient is getting closer to 4.
This suggests that the gradient of the curve y = x2 at the point (2, 4) is 4. x y 2 4
y = x2

9. The gradient of the chord is
Example (1)
Find the gradient of the chord joining the two points with x-coordinates 1 and on the graph of y = x2. Make a guess about the gradient of the tangent at the point x = 1.
The gradient of the chord is
(1.001, )
1.0012
= 2.001
I’d guess 2.
(1, ) 1)

10. The gradient of the chord is
Example (2)
Find the gradient of the chord joining the two points with x-coordinates 8 and on the graph of y = x2. Make a guess about the gradient of the tangent at the point x = 8.
The gradient of the chord is
(8.0001, ) =
I’d guess 16.
(8, ) 64

11. Let’s make a table of the results so far:
x-coordinate
gradient 1 2 4 3 5 6 7 8 16
You’re probably noticing a pattern here. But can we prove it mathematically?

12. I will call it ∆x.
I need to consider what happens when I increase x by a general increment. I will call it h.
(2 + h, (2 + h)2)
(2, 4) h

13. Let y = x2 and let A be the point (2, 4)
Let B be the point (2 + h, (2 + h)2) x y
A(2, 4)
y = x2 O
Here we have increased x by a very small amount h. In the early days of calculus h was referred to as an infinitesimal.
B(2 + h, (2 + h)2)
Draw the chord AB.
Gradient of AB
If h ≠ 0 we can cancel the h’s.
= 4 + h
Use a similar method to find the gradient of y = x2 at the points
(i) (3, 9)
(ii) (4, 16)
As h approaches zero, 4 + h approaches 4.
So the gradient of the curve at the point (2, 4) is 4.

14. We can now add to our table:
x-coordinate
gradient 1 2 4 3 5 6 7 8 16
It looks like the gradient is simply 2x. 6 8

15. Let’s check this result.
y = x2

16. Let’s check this result.
y = x2
Gradient at (3, 9) = 6

17. Let’s check this result.
y = x2
Gradient at (2, 4) = 4

18. Let’s check this result.
y = x2
Gradient at (1, 1) = 2

19. Let’s check this result.
y = x2
Gradient at (0, 0) = 0

20. Let’s check this result.
y = x2
Gradient at (–1, 1) = –2

21. Let’s check this result.
y = x2
Gradient at (–2, 4) = –4

22. Let’s check this result.
y = x2
Gradient at (–3, 9) = –6

23. Another way of seeing what the gradient is at the point (2, 4) is to plot an accurate graph and ‘zoom in’.
y = x2
ZOOM IN

24. When we zoom in the curve starts to look like a straight
line which makes it easy to
estimate the gradient.
0.8
0.2