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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
"Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"
The Rule for Differentiation
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
The Rule for Differentiation
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 Rule for Differentiation
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
The Rule for Differentiation
),( 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
The Rule for Differentiation
2xy 6m
The gradient changes as we move along a curve
e.g.
The Rule for Differentiation
2xy
4m
The Rule for Differentiation
2xy
2m
The Rule for Differentiation
2xy
0m
The Rule for Differentiation
2xy
2m
The Rule for Differentiation
2xy
4m
The Rule for Differentiation
2xy
6m
The Rule for Differentiation
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
The Rule for Differentiation
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
The Rule for Differentiation
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
The Rule for Differentiation
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
The Rule for Differentiation
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
The Rule for Differentiation
3xy Gradient of
dx
dy
)27,3( x
y3xy
The Rule for Differentiation
3xy Gradient of
3xy
)12,2( x
)27,3( xdx
dy
y
The Rule for Differentiation
3xy Gradient of
)27,3( x
)12,2( x
)3,1( x
3xy
dx
dy
y
The Rule for Differentiation
3xy Gradient of
)27,3( xdx
dy
)12,2( x
)3,1( x)0,0( x
3xy y
The Rule for Differentiation
3xy Gradient of
dx
dy
)27,3( x
)12,2( x
)3,1( x)0,0( x )3,1(x
3xy y
The Rule for Differentiation
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
The Rule for Differentiation
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
The Rule for Differentiation
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
The Rule for Differentiation
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
The Rule for Differentiation
34x
23x
45x
x2
1
dx
dy
3x4x5x
2x
x
y
1 0
Summary of Gradient Functions:
The Rule for Differentiation
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
The Rule for Differentiation
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 Rule for Differentiation
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
The Rule for Differentiation
3xy
tangent at x =
1
More Gradient Functions
gradient =
3
e.g. Multiplying by 2 multiplies the gradient by 2
3xy
The Rule for Differentiation
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
The Rule for Differentiation
e.g. 32xy 232 x
dx
dy
26x
Multiplying by a constant, multiplies the gradient by that constant
nx
The Rule for Differentiation
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
The Rule for Differentiation
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”
The Rule for Differentiation
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
The Rule for Differentiation
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 Rule for Differentiation
The Rule for Differentiation
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”
The Rule for Differentiation
e.g. 32xy 232 x
dx
dy
26x
e.g. 3752
1 23 xxxy
dx
dy
7102
3 2 xx
232
1x x25 7
The Rule for Differentiation
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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