PROGRAMME F11
DIFFERENTIATION
= slope finding
STROUD
Worked examples and exercises are in the text
Programme F11: Differentiation
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
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
Second derivatives
Newton-Raphson iterative method [optional]
STROUD
Worked examples and exercises are in the text
Programme F11: Differentiation
The gradient of a straight-line graph
The gradient of the sloping line straight line in the figure is defined as:
the vertical distance the line rises and falls between the two points P and Q
the horizontal distance between P and Q
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Worked examples and exercises are in the text
Programme F11: Differentiation
The gradient (slope) of a straight-line graph
The gradient of the sloping straight line in the figure is given as:
dy
and its value is denoted by the symbol m
dx
STROUD
Worked examples and exercises are in the text
Programme F11: Differentiation
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
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
Second derivatives
Newton-Raphson iterative method
STROUD
Worked examples and exercises are in the text
Programme F11: Differentiation
The AVERAGE gradient of a curve IN A REGION AROUND a given
point P
What you could call the average gradient of a curve between two points P
and Q will depend on the points chosen:
STROUD
Worked examples and exercises are in the text
The gradient of a curve AT a given point
The gradient of a curve at a point P is defined to be the gradient of the
tangent at that point [= the straight line that intersects the curve only at P,
when the curve is not itself a straight line around P - JAB ]:
NOTE: If the curve is a straight line around P, the tangent is just that line.
QUESTION: Does a graph always have well-defined tangent at a given point?? Consider
e.g. some graphs you’ve drawn in exercises involving the floor function, etc. [JAB]
STROUD
Worked examples and exercises are in the text
Programme F11: Differentiation
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
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
(Second derivatives –MOVED to a later set of slides)
Newton-Raphson iterative method
STROUD
Worked examples and exercises are in the text
Programme F11: Differentiation
Algebraic determination of the gradient of a curve
The gradient of the chord PQ is
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 y and the gradient of the tangent at P is dy
dx
x
Worked examples and exercises are in the text
Programme F11: Differentiation
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.
STROUD
Worked examples and exercises are in the text
Programme F11: Differentiation
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
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
Newton-Raphson iterative method
STROUD
Worked examples and exercises are in the text
Programme F11: Differentiation
Derivatives of powers of x
Two straight lines
Two curves
STROUD
Worked examples and exercises are in the text
Programme F11: Differentiation
Derivatives of powers of x
Two straight lines
(a) y  c (constant)
dy  0 therefore
STROUD
dy
0
dx
Worked examples and exercises are in the text
Programme F11: Differentiation
Derivatives of powers of x
Two straight lines
y  ax
(b)
y  dy  a( x  dx)
dy  a.dx therefore
dy
a
dx
QUESTION: what about a vertical line, x = d ?? [JAB]
STROUD
Worked examples and exercises are in the text
Programme F11: Differentiation
General definition of the derivative
dy/dx = limit of
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y/x
as  x  0
(from either side)
Worked examples and exercises are in the text
Programme F11: Differentiation
Derivatives of powers of x
Two curves
y  x2
(a)
y   y  ( x   x) 2
so
 y  2 x. x   x  therefore
2
therefore
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y
 2x   x
x
dy
 2x
dx
Worked examples and exercises are in the text
Programme F11: Differentiation
Derivatives of powers of x
Two curves
y  x3
(b)
y   y  ( x   x )3
so
 y  3x 2 . x  3x. x    x 
2
therefore
y
2
 3x 2  3x. x   x 
x
therefore
STROUD
3
dy
 3x 2
dx
Worked examples and exercises are in the text
Derivatives of powers of x
A clear pattern is emerging:
If y  x n then
dy
 nx n1
dx
EXERCISE: Prove this general result, using a result about (a+b)n that we
saw when studying combinations. [JAB]
STROUD
Worked examples and exercises are in the text
Algebraic determination of the gradient of a curve
y = 2x2 + 5
y   y  2 x   x  5
2
At Q:
 2 x 2  4 x. x  2  x   5
2
So
 y  4 x. x  2 x  and
2
As
 x  0 so
Therefore
y
 4 x  2. x
x
y
dy
 the gradient of the tangent at P 
x
dx
dy
 4x
dx
called the derivative of y with respect to x.
STROUD
Worked examples and exercises are in the text
Programme F11: Differentiation
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
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
Newton-Raphson iterative method
STROUD
Worked examples and exercises are in the text
Programme F11: Differentiation
Differentiation of polynomials
To differentiate a polynomial, we differentiate each term in turn:
If y  x 4  5 x 3  4 x 2  7 x  2
then
dy
 4 x3  5  3x 2  4  2 x  7 1  0
dx
Therefore
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dy
 4 x 3  15 x 2  8 x  7
dx
Worked examples and exercises are in the text
Programme F11: Differentiation
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
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
Newton-Raphson iterative method
STROUD
Worked examples and exercises are in the text
Programme F11: Differentiation
Derivatives – an alternative notation
The double statement:
If y  x 4  5 x 3  4 x 2  7 x  2
dy
then
 4 x3  5  3x 2  4  2 x  7 1  0
dx
can be written as:
d 4
x  5 x 3  4 x 2  7 x  2   4 x 3  15 x 2  8 x  7

dx
STROUD
Worked examples and exercises are in the text
Programme F11: Differentiation
Towards derivatives of trigonometric functions (JAB)
Limiting value of
sin 

as   0
is 1 [NB:  expressed in RADIANS]
[in lecture: a rough argument for this]
Following slide includes most of a rigorous argument.
STROUD
Worked examples and exercises are in the text
Programme F11: Differentiation
Area of triangle POA is:
1
2
r 2 sin 
Area of sector POA is:
1
2
r 2
Area of triangle POT is:
1
2
r 2 tan 
Therefore:
1
2
r 2 sin   12 r 2  12 r 2 tan  so 1 
That is ((using fact that the cosine tends to 1
Lim
 0
STROUD
sin 

sin 

 cos
-- JAB)):
1
Worked examples and exercises are in the text
Programme F11: Differentiation
Derivatives of trigonometric functions and …
The table of standard derivatives can be extended to include trigonometric
and the exponential functions:
d sin x
 cos x
dx
d cos x
  sin x
dx
de x
 ex
dx
[JAB:] The trig cases use the identities for finding sine and cosine of the sum of two
angles, and an approximation for the cosine of a small angle (in RADIANS):
cos x is approximately 1 – x2/2
STROUD
Worked examples and exercises are in the text
Programme F11: Differentiation
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
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
Newton-Raphson iterative method
STROUD
Worked examples and exercises are in the text
Programme F11: Differentiation
Differentiation of products of functions
Given the product of functions of x:
y  uv
then:
dy
dv
du
u v
dx
dx
dx
This is called the product rule.
STROUD
Worked examples and exercises are in the text
Programme F11: Differentiation
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
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
Newton-Raphson iterative method
STROUD
Worked examples and exercises are in the text
Programme F11: Differentiation
Differentiation of a quotient of two functions
Given the quotient of functions of x:
y
then:
u
v
du
dv
u
dy
 dx 2 dx
dx
v
v
This is called the quotient rule.
[BUT I find it easier to use the PRODUCT rule, replacing v by 1/v and using
the chain rule below. -- JAB]
STROUD
Worked examples and exercises are in the text
Programme F11: Differentiation
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
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function [i.e. compositions of functions]
Newton-Raphson iterative method
STROUD
Worked examples and exercises are in the text
Programme F11: Differentiation
Functions of a function (compositions)
To differentiate a composition w o u
we employ the chain rule.
If y is a function of u which is itself a function of x so that:
y = w(u(x))
e.g. y = sin (x2 + 1)
First, think of this as y = w(u),
or y = cos 2 x
e.g. y = sin u, with u = x2 + 1
dy dy du


dx du dx
Then:
This is called the chain rule.
STROUD
Worked examples and exercises are in the text
Programme F11: Differentiation
Compositions
Many functions of a function can be differentiated at sight by a slight
modification to the list of standard derivatives (F is the u of previous slide):
STROUD
Worked examples and exercises are in the text
Some Clarifications [JAB]
For any (differentiable) functions f(x) and g(x),
d/dx (f(x) + g(x)) = df(x)/dx + dg(x)/dx
d/dx (f(x) - g(x)) = df(x)/dx — dg(x)/dx
[and similarly for additions and subtractions of any number of functions]
d/dx kf(x) = k df(x)/dx where k is any constant.
d/dx xp
= p xp-1 where p is any non-zero constant (not just when it is a pos. integer)
d/dx (u/v) = d/dx (u.v -1) and you can deal with this by the product rule and the power
rule just above, instead of remembering the quotient rule separately.
STROUD
Worked examples and exercises are in the text
Programme F11: Differentiation
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
Differentiation of products of functions
Differentiation of a quotient of two functions
Functions of a function
Newton-Raphson iterative method [optional]
STROUD
Worked examples and exercises are in the text
Programme F11: Differentiation
Newton-Raphson iterative method [OPTIONAL]
Tabular display of results
Given that x0 is an approximate solution to the equation f(x) = 0 then a better
solution is given as x1, where:
x1  x0 
f ( x0 )
f ( x0 )
This gives rise to a series of improving solutions by iteration using:
xn 1  xn 
f ( xn )
f ( xn )
A tabular display of improving solutions can be produced in a spreadsheet.
STROUD
Worked examples and exercises are in the text