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© Boardworks Ltd 2005 1 of 45 © Boardworks Ltd 2005 1 of 45 AS-Level Maths: Core 1 for Edexcel C1.7 Differentiation This icon indicates the slide contains activities created in Flash. These activities are not editable. For more detailed instructions, see the Getting Started presentation.

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© Boardworks Ltd 2005 2 of 45 Contents © Boardworks Ltd 2005 2 of 45 Rates of change The gradient of the tangent at a point The gradient of the tangent as a limit Differentiation of polynomials Second order derivatives Tangents and normals Examination-style questions Rates of change

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© Boardworks Ltd 2005 3 of 45 Rates of change This graph shows the distance that a car travels over a period of 5 seconds. The gradient of the graph tells us the rate at which the distance changes with respect to time. In other words, the gradient tells us the speed of the car. The car in this example is travelling at a constant speed since the gradient is the same at every point on the graph. time (s) distance (m) 0 5 40

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© Boardworks Ltd 2005 4 of 45 Rates of change In most situations, however, the speed will not be constant and the distance–time graph will be curved. For example, this graph shows the distance–time graph as the car moves off from rest. The speed of the car, and therefore the gradient, changes as you move along the curve. To find the rate of change in speed we need to find the gradient of the curve. The process of finding the rate at which one variable changes with respect to another is called differentiation. time (s) distance (m) 0 In most situations this involves finding the gradient of a curve.

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© Boardworks Ltd 2005 5 of 45 Contents © Boardworks Ltd 2005 5 of 45 Rates of change The gradient of the tangent at a point The gradient of the tangent as a limit Differentiation of polynomials Second order derivatives Tangents and normals Examination-style questions The gradient of the tangent at a point

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© Boardworks Ltd 2005 6 of 45 The gradient of a curve The gradient of a curve at a point is given by the gradient of the tangent at that point. Look at how the gradient changes as we move along a curve:

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© Boardworks Ltd 2005 7 of 45 Finding the gradient of a curve at a point Suppose we want to find the gradient of the curve y = x 2 at the point (3, 9). Use graph paper to carefully plot the curve y = x 2 for –5 ≤ x ≤ 5. Use a ruler to draw the tangent at the point (3, 9) and find the gradient of the tangent. Find the gradients of the tangents at four other points and write down what you notice. As you can see, it is quite difficult to do this accurately because we only know a single point that the line passes through. To find the gradient of a line we need to know two points on the line ( x 1, y 1 ) and ( x 2, y 2 ). We can then find its gradient:

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© Boardworks Ltd 2005 8 of 45 Contents © Boardworks Ltd 2005 8 of 45 Rates of change The gradient of the tangent at a point The gradient of the tangent as a limit Differentiation of polynomials Second order derivatives Tangents and normals Examination-style questions The gradient of the tangent as a limit

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© Boardworks Ltd 2005 9 of 45 Differentiation from first principles Suppose we want to find the gradient of a curve at a point A. We can add another point B on the line close to point A. As point B moves closer to point A, the gradient of the chord AB gets closer to the gradient of the tangent at A. δx represents a small change in x and δy represents a small change in y.

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© Boardworks Ltd 2005 10 of 45 Differentiation from first principles We can write the gradient of the chord AB as: As B gets closer to A, δx gets closer to 0 and gets closer to the value of the gradient of the tangent at A. δx can’t actually be equal to 0 because we would then have division by 0 and the gradient would then be undefined. Instead we must consider the limit as δx tends to 0. This means that δx becomes infinitesimal without actually becoming 0.

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© Boardworks Ltd 2005 11 of 45 Differentiation from first principles Let’s see how this works for the gradient of the tangent to the curve y = x 2 at the point A (3, 9).

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© Boardworks Ltd 2005 12 of 45 Differentiation from first principles If A is the point (3, 9) on the curve y = x 2 and B is another point close to (3, 9) on the curve, we can write the coordinates of B as (3 + δx, (3 + δx ) 2 ). The gradient of chord AB is: δxδx δyδy A (3, 9) B (3 + δx, (3 + δx ) 2 )

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© Boardworks Ltd 2005 13 of 45 Differentiation from first principles We write this as: Let’s apply this method to a general point on the curve y = x 2. If we let the x -coordinate of a general point A on the curve y = x 2 be x, then the y -coordinate will be x 2. At the limit where δx → 0, 6 + δx → 6. 6 So the gradient of the tangent to the curve y = x 2 at the point (3, 9) is 6. So, A is the point ( x, x 2 ). If B is another point close to A ( x, x 2 ) on the curve, we can write the coordinates of B as ( x + δx, ( x + δx ) 2 ).

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© Boardworks Ltd 2005 14 of 45 Differentiation from first principles The gradient of chord AB is: δxδx δyδy A ( x, x 2 ) B ( x + δx, ( x + δx ) 2 )

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© Boardworks Ltd 2005 15 of 45 The gradient of y = x 2

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© Boardworks Ltd 2005 16 of 45 The gradient function So the gradient of the tangent to the curve y = x 2 at the general point ( x, y ) is 2 x. 2 x is often called the gradient function or the derived function of y = x 2. If the curve is written using function notation as y = f ( x ), then the derived function can be written as f ′( x ). So, if: f ( x ) = x 2 This notation is useful if we want to find the gradient of f ( x ) at a particular point. For example, the gradient of f ( x ) = x 2 at the point (5, 25) is: f ′(5) = 2 × 5 =10 f ′( x ) = 2 x Then:

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© Boardworks Ltd 2005 17 of 45 The gradient function of y = x 2

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© Boardworks Ltd 2005 18 of 45 Differentiating y = x 3 from first principles For the curve y = x 3, we can consider the general point A ( x, x 3 ) and the point B ( x + δx, ( x + δx ) 3 ) a small distance away from A. The gradient of chord AB is then:

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© Boardworks Ltd 2005 19 of 45 The gradient of y = x 3

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© Boardworks Ltd 2005 20 of 45 The gradient of y = x 3 So, the gradient of the tangent to the curve y = x 3 at the general point ( x, y ) is 3 x 2. So if, f ( x ) = x 3 What is the gradient of the tangent to the curve f ( x ) = x 3 at the point (–2, –8)? f ′( x ) = 3 x 2 f ′(–2) = 3(–2) 2 =12 Explain why the gradient of f ( x ) = x 3 will always be positive. f ′( x ) = 3 x 2 Then:

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© Boardworks Ltd 2005 21 of 45 The gradient function of y = x 3

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© Boardworks Ltd 2005 22 of 45 Using the notation We have shown that for y = x 3 dy dx represents the derivative of y with respect to x. is usually written as. So if y = x 3 of the tangent. Remember, is the gradient of a chord, while is the gradient then:

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© Boardworks Ltd 2005 23 of 45 Using the notation dy dx This notation can be adapted for other variables so, for example: Also, if we want to differentiate 2 x 4 with respect to x, for example, we can write: represents the derivative of s with respect to t. We could work this out by differentiating from first principles, but in practice this is unusual. If s is distance and t is time then we can interpret this as the rate of change in distance with respect to time. In other words, the speed.

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© Boardworks Ltd 2005 24 of 45 Contents © Boardworks Ltd 2005 24 of 45 Rates of change The gradient of the tangent at a point The gradient of the tangent as a limit Differentiation of polynomials Second order derivatives Tangents and normals Examination-style questions Differentiation of polynomials

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© Boardworks Ltd 2005 25 of 45 Differentiation If we continued the process of differentiating from first principles we would obtain the following results: yxx2x2 x3x3 x4x4 x5x5 x6x6 What pattern do you notice? In general: and when x n is preceded by a constant multiplier a we have: 12x2x 3x23x2 4x34x3 5x45x4 6x56x5

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© Boardworks Ltd 2005 26 of 45 y x 0 Differentiation For example, if y = 2 x 4 Suppose we have a function of the form y = c, where c is a constant number. This corresponds to a horizontal line through the point (0, c). c The gradient of this line is always equal to 0. If y = c

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© Boardworks Ltd 2005 27 of 45 Differentiation Find the gradient of the curve y = 3 x 4 at the point (–2, 48). Differentiating: At the point (–2, 48) x = –2 so: The gradient of the curve y = 3 x 4 at the point (–2, 48) is –96.

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© Boardworks Ltd 2005 28 of 45 Differentiating polynomials Suppose we want to differentiate a polynomial function. For example: Differentiate y = x 4 + 3 x 2 – 5 x + 2 with respect to x. In general, if y is made up of the sum or difference of any given number of functions, its derivative will be made up of the derivative of each of the functions. We can differentiate each term in the function to give. 4x34x3 + 6 x – 5 So:

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© Boardworks Ltd 2005 29 of 45 Differentiating polynomials Find the point on y = 4 x 2 – x – 5 where the gradient is 15. when 8 x – 1 = 15 8 x = 16 x = 2 We now substitute this value into the equation y = 4 x 2 – x – 5 to find the value of y when x = 2 y = 4(2) 2 – 2 – 5 The gradient of y = 4 x 2 – x – 5 is 15 at the point (2, 9). = 9

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© Boardworks Ltd 2005 30 of 45 Differentiating polynomials In some cases, a function will have to be written as separate terms containing powers of x before differentiating. For example: Given that f ( x ) = (2 x – 3 ) ( x 2 – 5) find f ′( x ). Expanding: f ( x ) = 2 x 3 – 3 x 2 – 10 x + 15 f ′( x ) = 6 x 2 – 6 x –10 Find the gradient of f ( x ) at the point (–3, –36). When x = –3 we have f ′(–3) = 6(–3) 2 – 6(–3) –10 = 54 + 18 – 10 = 62

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© Boardworks Ltd 2005 31 of 45 Differentiating polynomials = 6 x 3 + 2 We can now differentiate: = 2(3 x 3 + 1)

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© Boardworks Ltd 2005 32 of 45 Differentiating ax n for all rational n This is true for all negative or fractional values of n. For example: Start by writing this as y = 2 x –1 So:

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© Boardworks Ltd 2005 33 of 45 Differentiating ax n for all rational n So: Start by writing this as Remember, in some cases, a function will have to be written as separate terms containing powers of x before differentiating.

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© Boardworks Ltd 2005 34 of 45 Differentiating ax n for all rational n

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© Boardworks Ltd 2005 35 of 45 Contents © Boardworks Ltd 2005 35 of 45 Rates of change The gradient of the tangent at a point The gradient of the tangent as a limit Differentiation of polynomials Second order derivatives Tangents and normals Examination-style questions Second order derivatives

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© Boardworks Ltd 2005 36 of 45 Second order derivatives Differentiating a function y = f ( x ) gives us the derivative or f ′ ( x ) Differentiating the function a second time gives us the second order derivative. This can be written as or f ′′ ( x ) The second order derivative gives us the rate of change of the gradient of a function. We can think of it as the gradient of the gradient.

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© Boardworks Ltd 2005 37 of 45 Using second order derivatives

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© Boardworks Ltd 2005 38 of 45 Contents © Boardworks Ltd 2005 38 of 45 Rates of change The gradient of the tangent at a point The gradient of the tangent as a limit Differentiation of polynomials Second order derivatives Tangents and normals Examination-style questions Tangents and normals

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© Boardworks Ltd 2005 39 of 45 Tangents and normals Remember, the tangent to a curve at a point is a straight line that just touches the curve at that point. The normal to a curve at a point is a straight line that is perpendicular to the tangent at that point. We can use differentiation to find the equation of the tangent or the normal to a curve at a given point. For example: Find the equation of the tangent and the normal to the curve y = x 2 – 5 x + 8 at the point P(3, 2).

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© Boardworks Ltd 2005 40 of 45 Tangents and normals y = x 2 – 5 x + 8 At the point P(3, 2) x = 3 so: 1 The gradient of the tangent at P is therefore 1. Using y – y 1 = m ( x – x 1 ), give the equation of the tangent at the point P(3, 2): y – 2 = x – 3 y = x – 1

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© Boardworks Ltd 2005 41 of 45 Tangents and normals The normal to the curve at the point P(3, 2) is perpendicular to the tangent at that point. Using y – y 1 = m ( x – x 1 ) give the equation of the tangent at the point P(3, 2): The gradient of the tangent at P is 1 and so the gradient of the normal is –1.

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© Boardworks Ltd 2005 42 of 45 Contents © Boardworks Ltd 2005 42 of 45 Rates of change The gradient of the tangent at a point The gradient of the tangent as a limit Differentiation of polynomials Second order derivatives Tangents and normals Examination-style questions

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© Boardworks Ltd 2005 43 of 45 Examination-style question The curve f ( x ) = x 2 + 2 x – 8 cuts the x -axes at the points A(– a, 0) and B( b, 0) where a and b are positive integers. a)Find the coordinates of points A and B. b)Find the gradient function of the curve. c)Find the equations of the normals to the curve at points A and B. d)Given that these normals intersect at the point C, find the coordinates of C. a) The equation of the curve can be written in factorized form as y = ( x + 4)( x – 2) so the coordinates of A are (–4, 0) and the coordinates of B are (2, 0).

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© Boardworks Ltd 2005 44 of 45 Examination-style question b) f ( x ) = x 2 + 2 x – 8 f’ ( x ) = 2 x + 2 c) At (–4, 0) the gradient of the curve is –8 + 2 = –6 Using y – y 1 = m ( x – x 1 ), the equation of the normal at (–4, 0) is: The gradient of the normal at (–4, 0) is. y – 0 = ( x + 4) 1 6 y = x + 4 At (2, 0) the gradient of the curve is 4 + 2 = 6 Using y – y 1 = m ( x – x 1 ), the equation of the normal at (2, 0) is: The gradient of the normal at (2, 0) is –. y – 0 = – ( x – 2) 2 6 y = 2 – x

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© Boardworks Ltd 2005 45 of 45 Examination-style question x + 4 = 2 – x d) Equating and : 12 2 x = –2 x = 1 When x = –1, y =. So the coordinates of C are (–1, ).

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