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**“Teach A Level Maths” Vol. 1: AS Core Modules**

15: The Gradient of the Tangent as a Limit © Christine Crisp

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**Module C1 Module C2 AQA MEI/OCR Edexcel OCR**

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**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.

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**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 ).

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**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.

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**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

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**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.

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**We can get an even better estimate if we use the point .**

Tangent at A Chord AB3 The gradient of AB3 is

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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.

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**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 ”

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