1. “Teach A Level Maths” Vol. 1: AS Core Modules
15: The Gradient of the Tangent as a Limit
© Christine Crisp

2. Module C1 Module C2 AQA MEI/OCR Edexcel OCR
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3. The Gradient of a Tangent
We found the rule for differentiating by noticing a pattern in results found by measuring gradients of tangents.
However, if we want to prove the rule or find a rule for some other functions we need a method based on algebra.
This presentation shows you how this is done.
The emphasis in this presentation is upon understanding ideas rather than doing calculations.

4. Consider the tangent at the point A( 1, 1 ) on
Tangent at A
A(1,1)
(2,4)
As an approximation to the gradient of the tangent we can use the gradient of a chord from A to a point close to A.
e.g. we can use the chord to the point ( 2, 4 ).
( We are going to use several points, so we’ll call this point B1 ).

5. We can see this gradient is larger than the gradient of the tangent.
Consider the tangent at the point A( 1, 1 ) on
(2,4) B1
Chord AB1
Tangent at A
A(1,1)
The gradient of the chord AB1 is given by
We can see this gradient is larger than the gradient of the tangent.

6. The gradient of the chord AB2 is
To get a better estimate we can take a point B2 that is closer to A( 1, 1 ), e.g.
A(1,1)
Tangent at A B1
Chord AB2
The gradient of the chord AB2 is

7. We can get an even better estimate if we use the point .
Tangent at A
Chord AB3
We need to zoom in to the curve to see more clearly.

8. We can get an even better estimate if we use the point .
Tangent at A
Chord AB3
The gradient of AB3 is

9. Continuing in this way, moving B closer and closer to A( 1, 1 ), and collecting the results in a table, we get x
y - 1
x - 1
Point
Gradient of AB
As B gets closer to A, the gradient approaches 2. This is the gradient of the tangent at A.

10. As B gets closer to A, the gradient of the chord AB approaches the gradient of the tangent.
We write that the gradient of the tangent at A
The gradient of the tangent at A is “ the limit of the gradient of the chord AB as B approaches A ”

11. We will generalize the result above to find a formula for the gradient at any point on a curve given by
STUDENTS!
If you are working through this on your own, ask your teacher if you need to do the next ( final ) section.

12. Let A be the point ( x, ) on the curve
We need a general notation for the coordinates of B that suggests it is near to A.
We use
is the Greek letter d so we can think of as standing for “difference”.
So, is the small difference in x as we move from A to B.
can be used for the difference in y values.

13. We can’t put x = 0 in this as we would get which is undefined.
We have
and
So, the gradient of the chord AB is m where
So, since the
gradient of the tangent =
the gradient of the tangent =
We can’t put x = 0 in this as we would get which is undefined.

14. the gradient of the tangent =
( the letter h is sometimes used instead of )
If we use for the difference in y-values,
the gradient of the tangent =
We then get
But, the gradient of the tangent gives the gradient of the curve, so

15. e.g. Prove that the gradient function of the curve
where is given by
Solution: The gradient of the curve is given by the gradient of the tangent, so
For ,
So,
Since has cancelled, we will not be dividing by zero.
So, as ,

17. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.
For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

18. The gradient of the chord AB1 is given by
Consider the tangent at the point A( 1, 1 ) on
A(1,1)
(2,4)
Tangent at A
We can see this gradient is larger than the gradient of the tangent.

19. We write that the gradient of the tangent at A
As B gets closer to A, the gradient of the chord AB approaches the gradient of the tangent.
We will generalize the result above to find a formula for the gradient at any point on a curve given by
The gradient of the tangent at A is “ the limit of the gradient of the chord AB as B approaches A ”

20. We need a general notation for the coordinates of B that suggests it is near to A.
So, is the small difference in x as we move from A to B.
is the Greek letter d so we can think of as standing for “difference”.
Let A be the point ( x, ) on the curve
We use
can be used for the difference in y values.

21. We can’t put x = 0 in this as we would get which is undefined.
We have
and
So, the gradient of the chord AB is m where
So, since the
gradient of the tangent =
the gradient of the tangent =
We can’t put x = 0 in this as we would get which is undefined.

22. the gradient of the tangent =
But, the gradient of the tangent gives the gradient of the curve, so
If we use for the difference in y-values,
( the letter h is sometimes used instead of )
We then get

23. Solution: The gradient of the curve is given by the gradient of the tangent, so
So, as ,
e.g. Prove that the gradient function of the curve
where is given by
For ,
So,
Since has cancelled, we will not be dividing by zero.