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 ”