Download - 12 Differentiation
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Differentiation
M. Norazizi Sham Mohd Sayuti, PhD
Faculty Science and Technology
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STROUD Worked examples and exercises are in the text
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The gradient of a straight-line graph
The gradient of a curve at a given point
Algebraic determination of the gradient of a curve
Derivatives of powers of x
Differentiation of polynomials
Derivatives – an alternative notation
Standard derivatives and rules
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
2
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STROUD Worked examples and exercises are in the text
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The gradient of a straight-line graph
The gradient of the sloping straight line in the figure is defined as:
the vertical distance the line rises and falls between the two points P and Qthe horizontal distance between P and Q
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STROUD Worked examples and exercises are in the text
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Math
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The gradient of a straight-line graph
The gradient of a curve at a given point
Algebraic determination of the gradient of a curve
Derivatives of powers of x
Differentiation of polynomials
Derivatives – an alternative notation
Standard derivatives and rules
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
5
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STROUD Worked examples and exercises are in the text
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The gradient of a curve at a given point
If we take two points P and Q on a curve and calculate, as we did for the straight line, the ratio of the vertical distance the curve rises or falls and the horizontal distance between P and Q the result will depend on the points chosen:
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STROUD Worked examples and exercises are in the text
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The gradient of a curve at a point P is defined to be the gradient of the tangent at that point:
The gradient of a curve at a given point
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STROUD Worked examples and exercises are in the text
KEH
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Math
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The gradient of a straight-line graph
The gradient of a curve at a given point
Algebraic determination of the gradient of a curve
Derivatives of powers of x
Differentiation of polynomials
Derivatives – an alternative notation
Standard derivatives and rules
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
9
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STROUD Worked examples and exercises are in the text
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Algebraic determination of the gradient of a curve
The gradient of the chord PQ is and the gradient of the tangent at P is y
x
dy
dx
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STROUD Worked examples and exercises are in the text
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Algebraic determination of the gradient of a curve
As Q moves to P so the chord rotates. When Q reaches P the chord is coincident with the tangent.
For example, consider the graph of 22 5y x=
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STROUD Worked examples and exercises are in the text
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Algebraic determination of the gradient of a curve
At Q:
So
As
Therefore
called the derivative of y with respect to x.
( )2
2 5y y x x =
222 4 . 2 5x x x x =
2
4 . 2 and 4 2.y
y x x x x xx
= =
0 so the gradient of the tangent at y dy
x Px dx
=
4dy
xdx
=
12
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STROUD Worked examples and exercises are in the text
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Math
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The gradient of a straight-line graph
The gradient of a curve at a given point
Algebraic determination of the gradient of a curve
Derivatives of powers of x
Differentiation of polynomials
Derivatives – an alternative notation
Standard derivatives and rules
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
13
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STROUD Worked examples and exercises are in the text
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Derivatives of powers of x
Two straight lines
(a) (constant)y c=
0 therefore 0dy
dydx
= =
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STROUD Worked examples and exercises are in the text
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Derivatives of powers of x
Two straight lines
(b) y ax=
. therefore dy
dy a dx adx
= =
( ) y dy a x dx =
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STROUD Worked examples and exercises are in the text
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Derivatives of powers of x
Two curves
(a)
so
2 y x=
therefore 2dy
xdx
=
2( ) y y x x =
2
2 . therefore 2y
y x x x x xx
= =
16
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STROUD Worked examples and exercises are in the text
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Derivatives of powers of x
A clear pattern is emerging:
1If then n ndyy x nx
dx-= =
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STROUD Worked examples and exercises are in the text
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Math
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The gradient of a straight-line graph
The gradient of a curve at a given point
Algebraic determination of the gradient of a curve
Derivatives of powers of x
Differentiation of polynomials
Derivatives – an alternative notation
Standard derivatives and rules
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
19
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STROUD Worked examples and exercises are in the text
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Differentiation of polynomials
To differentiate a polynomial, we differentiate each term in turn:
4 3 2
3 2
3 2
If 5 4 7 2
then 4 5 3 4 2 7 1 0
Therefore 4 15 8 7
y x x x x
dyx x x
dx
dyx x x
dx
= - -
= - -
= -
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STROUD Worked examples and exercises are in the text
KEH
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Math
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The gradient of a straight-line graph
The gradient of a curve at a given point
Algebraic determination of the gradient of a curve
Derivatives of powers of x
Differentiation of polynomials
Derivatives – an alternative notation
Standard derivatives and rules
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
24
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STROUD Worked examples and exercises are in the text
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Standard derivatives and rules
Standard derivatives
The table of standard derivatives can be extended to include trigonometric and the exponential functions:
d
dxsin x( ) = cos x
d
dxcos x( ) = -sin x
d
dxex( ) = ex
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STROUD Worked examples and exercises are in the text
KEH
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Math
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The gradient of a straight-line graph
The gradient of a curve at a given point
Algebraic determination of the gradient of a curve
Derivatives of powers of x
Differentiation of polynomials
Derivatives – an alternative notation
Standard derivatives and rules
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
28
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STROUD Worked examples and exercises are in the text
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Differentiation of products of functions
Given the product of functions of x:
then:
This is called the product rule.
y uv=
dy dv duu v
dx dx dx=
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STROUD Worked examples and exercises are in the text
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Differentiation of products of functions
If:
then
For example:
y uv
dy dv duu v
dx dx dx
=
=
( )
3
3 2
2
.sin 3 then
.3cos3 3 sin 3
3 cos3 sin 3
y x x
dyx x x x
dx
x x x x
=
=
=
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STROUD Worked examples and exercises are in the text
KEH
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Math
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The gradient of a straight-line graph
The gradient of a curve at a given point
Algebraic determination of the gradient of a curve
Derivatives of powers of x
Differentiation of polynomials
Derivatives – an alternative notation
Standard derivatives and rules
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
31
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STROUD Worked examples and exercises are in the text
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Differentiation of a quotient of two functions
Given the quotient of functions of x:
then:
This is called the quotient rule.
uy
v=
2
du dvv u
dy dx dxdx v
-=
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STROUD Worked examples and exercises are in the text
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Differentiation of a quotient of two functions
If:
then
For example:
2
u
yv
du dvv u
dy dx dxdx v
=
-=
2
sin 3 then
1
( 1)3cos3 sin 3 .1
( 1)
xy
x
dy x x x
dx x
=
-=
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STROUD Worked examples and exercises are in the text
KEH
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Math
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The gradient of a straight-line graph
The gradient of a curve at a given point
Algebraic determination of the gradient of a curve
Derivatives of powers of x
Differentiation of polynomials
Derivatives – an alternative notation
Standard derivatives and rules
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
34
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STROUD Worked examples and exercises are in the text
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Functions of a function
Differentiation of a function of a function
To differentiate a function of a function we employ the chain rule.
If y is a function of u which is itself a function of x so that:
Then:
This is called the chain rule.
( ) ( [ ])y x y u x=
dy dy du
dx du dx=
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Functions of a function
If:
then
For example:
( ) and ( )
. (called the 'chain rule')
y f u u F x
dy dy du
dx du dx
= =
=
cos(5 4) so cos and 5 4
.
( sin ).5
5sin(5 4)
y x y u u x
dy dy du
dx du dx
u
x
= - = = -
=
= -
= - -36