11: The Rule for Differentiation © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules

11: 11: The Rule for The Rule for

DifferentiationDifferentiation

© Christine Crisp

““Teach A Level Maths”Teach A Level Maths”

Vol. 1: AS Core Vol. 1: AS Core ModulesModules

The Rule for Differentiation

Module C1AQA

EdexcelOCR

MEI/OCR

Module C2

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The gradient of a straight line is given by

The Gradient of a Straight Line

12

12

xx

yym

where and are points on the line ),( 11 yx ),( 22 yx


7 - 1 = 6

Solution:

3 - 1 = 2

difference in the x-values

difference in the y-values

12

12

xx

yym

32

6

13

17

m

)7,3(x

)1,1(x

e.g. Find the gradient of the line joining the points with coordinates and)1,1( )7,3(


The gradient of a straight line is given by

We use this idea to get the gradient at a point on a curve

This branch of Mathematics is called Calculus

Gradients are important as they measure the rate of change of one variable with another. For the graphs in this section, the gradient measures how y changes with x

values the in difference the

values- the in difference the

xy

m


),( 42Tangent at

2xy (2, 4)x

The Gradient at a point on a CurveDefinition: The gradient of a point on a curve

equals the gradient of the tangent at that point.e.g.

3

12

The gradient of the tangent at (2, 4) is

43

12 mSo, the gradient of the curve at (2, 4)

is 4


2xy 6m

The gradient changes as we move along a curve

e.g.


2xy

4m


2xy

2m


2xy

0m


2xy

2m


2xy

4m


2xy

6m


For the curve we have the following gradients:

2xy Point on the curve

Gradient

)4,2(

)9,3(

)1,1(

)0,0(

)1,1(

)4,2(

)9,3(

64

2

0

2

4

6

At every point, the gradient is twice the x-value


Point on the curve

For the curve we have the following gradients:

2xy

Gradient

64

2

0

2

4

6

)4,(

)9,(

)1,(

)0,0(

)1,(

)4,(

)9,(

21

1

2

3

3

At every point, the gradient is twice the x-value


At every point on the gradient is twice the x-value

2xy

This rule can be written as xdx

dy2

The notation comes from the idea of the gradient of a line being

the iff erence in the - valuesthe iff erence in the - valuesd

dxy


d

d xy

At every point on the gradient is twice the x-value

2xy

This rule can be written as xdx

dy2

The notation comes from the idea of the gradient of a line being

is read as “ dy by dx ”dx

dy

The function giving the gradient of a curve is called the gradient function

the iff erence in the - valuesthe iff erence in the - values


34xdx

dy4xy

45xdx

dy5xy

The rule for the gradient function of a curve of the form nxy 1 nnx

dx

dyis

23xdx

dy3xy

“subtract 1 from the power”

“power to the front and multiply”

Although this rule won’t be proved, we can illustrate it for by sketching the gradients at points on the curve

3xy

Other curves and their gradient functions


3xy Gradient of

dx

dy

)27,3( x

y3xy


3xy Gradient of

3xy

)12,2( x

)27,3( xdx

dy

y


3xy Gradient of

)27,3( x

)12,2( x

)3,1( x

3xy

dx

dy

y


3xy Gradient of

)27,3( xdx

dy

)12,2( x

)3,1( x)0,0( x

3xy y


3xy Gradient of

dx

dy

)27,3( x

)12,2( x

)3,1( x)0,0( x )3,1(x

3xy y


3xy Gradient of

dx

dy

)27,3( x

)12,2( x

)3,1( x)0,0( x )3,1(x

)12,2(x

3xy y


3xy Gradient of

dx

dy

)27,3( x

)12,2( x

)3,1( x)0,0( x )3,1(x

)12,2(x

)27,3(x

3xy y


23xdx

dy

3xy

dx

dy

)27,3( x

)12,2( x

)3,1( x)0,0( x )3,1(x

)12,2(x

)27,3(x

y


1yxy

The gradients of the functions and can also be found by the rule but as they represent straight lines we already know their gradients

xy 1y

gradient = 0

gradient = 1


34x

23x

45x

x2

1

dx

dy

3x4x5x

2x

x

y

1 0

Summary of Gradient Functions:


The Gradient Function and Gradient at a Point

e.g.1 Find the gradient of the curve at the point (2, 12).

3xy

dx

dy

At x = 2, the gradient

2)2(3dx

dym

Solution:

3xy

23x

12


Exercises

1. Find the gradient of the curve at the point where x 1

4xy

At x = - 1, m = -434xdx

dySolution

:

2. Find the gradient of the curve at the

point

3xy

81

21 ,

23xdx

dySolution

:

432

21

21 3, mxAt


The process of finding the gradient function is called differentiation.

The gradient function is called the derivative.

The rule for differentiating can be extended to curves of the form

where a is a constant.

naxy


3xy

tangent at x =

1

More Gradient Functions

gradient =

3

e.g. Multiplying by 2 multiplies the gradient by 2

3xy


32xy

gradient =

6

e.g. Multiplying by 2 multiplies the gradient by 2

3xy

3xy

tangent at x =

1

tangent at x =

1

gradient =

3


e.g. 32xy 232 x

dx

dy

26x

Multiplying by a constant, multiplies the gradient by that constant

nx


The rule can also be used for sums and differences of terms.

e.g. 3752

1 23 xxxy

dx

dy

7102

3 2 xx

232

1x x25 7


e.g. Find the gradient at the point where x = 1 on the curve

423 23 xxxySolution: Differentiating to find the gradient

function:

149 2 xxdx

dy

When x = 1, gradient m =

1)1(4)1(9 2

12 m

Using Gradient Functions

The Rule for DifferentiationSUMMARY The gradient at a point on a curve is

defined as the gradient of the tangent at that point

The process of finding the gradient function is called differentiating

The function that gives the gradient of a curve at any point is called the gradient function

The rule for differentiating terms of the form

1 nanxdx

dynaxy is

• “power to the front and multiply”• “subtract 1 from the power”


7106 2 xxdx

dy

2612 2 xxdx

dy

2xdx

dy

Find the gradients at the given points on the following curves:

1. 3752 23 xxxyat the point

)1,1(

2. 4234 23 xxxy at the point

)5,1(

3. 12221 xxy at the

point )1,2(

When 3,1 mx

4,1 mxWhen

0,2 mxWhen

Exercises


x

xxy

23 5.

4. )4)(2( xxy at )9,1( ( Multiply out the brackets before using the rule )

(Divide out before using the rule )

822 xxy

22 xy

22 xdx

dy

xdx

dy2

at )2,2(

x

x

x

xy

23

0,1 mxWhen

42 mxWhen

Find the gradients at the given points on the following curves:

Exercises



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.

The Rule for DifferentiationSUMMARY The gradient at a point on a curve is

defined as the gradient of the tangent at that point

The process of finding the gradient function is called differentiating

The function that gives the gradient of a curve at any point is called the gradient function

The rule for differentiating terms of the form

1 nanxdx

dynaxy is

• “power to the front and multiply”• “subtract 1 from the power”


e.g. 32xy 232 x

dx

dy

26x

e.g. 3752

1 23 xxxy

dx

dy

7102

3 2 xx

232

1x x25 7


e.g. Find the gradient at the point where x = 1 on the curve

423 23 xxxySolution: Differentiating to find the gradient

function:

149 2 xxdx

dy

When x = 1, gradient m =

1)1(4)1(9 2

12 m

Using Gradient Functions

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11: The Rule for Differentiation © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules