PROGRAMME 9
DIFFERENTIATION
APPLICATIONS 2
((edited by JAB))
STROUD
Worked examples and exercises are in the text
Programme 9: Differentiation applications 2
NOT IN TEST or EXAM but IMPORTANT
Differentiation of inverse trigonometric functions
Derivatives of inverse hyperbolic functions [REMOVED]
Second derivatives
Maximum and minimum values
Points of inflexion
STROUD
Worked examples and exercises are in the text
Programme 9: Differentiation applications 2
Differentiation of inverse trigonometric functions
Derivatives of inverse hyperbolic functions
Second derivatives
Maximum and minimum values
Points of inflexion
STROUD
Worked examples and exercises are in the text
(moved here from) Programme F11: Differentiation
Second derivatives
Notation
The derivative of the derivative of y is called the second derivative of y and
is written as:
d  dy  d 2 y
 
dx  dx  dx 2
So, if:
y  x 4  5 x3  4 x 2  7 x  2
dy
 4 x 3  15 x 2  8 x  7
dx
then
d2y
2

12
x
 30 x  8
2
dx
STROUD
Worked examples and exercises are in the text
Third etc. derivatives (added by JAB)
Can continue like that differentiating again and again.
E.g., the third derivative of y wrt x is notated as d3y/dx3
and is the derivative of d2 y/dx2
STROUD
Worked examples and exercises are in the text
Programme 9: Differentiation applications 2
Differentiation of inverse trigonometric functions
Derivatives of inverse hyperbolic functions
Second derivatives
Local maxima, local minima and points of inflexion (partial treatment)
STROUD
Worked examples and exercises are in the text
(slide added by JAB)
Local Maximum and Local Minima
LOCAL MAXIMUM at point P:
Small enough deviations to the left and right of P always make y
DECREASE.
LOCAL MINIMUM at point P:
Small enough deviations to the left and right of P always make y
INCREASE.
STROUD
Worked examples and exercises are in the text
Programme 9: Differentiation applications 2
Stationary points
A stationary point is a point on the
graph of a function y = f (x) where
the rate of change is zero. That is
where:
dy
0
dx
This can occur at a local maximum,
a local minimum or a point of
inflexion. Solving this equation will
locate the stationary points.
STROUD
Worked examples and exercises are in the text
Programme 9: Differentiation applications 2
Local maximum and minimum values
Having located a stationary point
one can characterize it further. If, at
the stationary point
d2y
0
2
dx
the stationary point is a minimum
d2y
0
dx 2
the stationary point is a maximum
CAUTION: Local maxima and minima can also
occur when the 2nd derivative is zero. E.g.,
consider
y = x4
STROUD
Worked examples and exercises are in the text
Programme 9: Differentiation applications 2
Maximum and minimum values
If, at the stationary point
d2y
0
2
dx
The stationary point may be:
a local maximum,
a local minimum or
a point of inflexion
The test is to look at the values of y a
little to the left and a little to the
right of the stationary point
STROUD
Worked examples and exercises are in the text
(slide added by JAB)
Maximum and minimum values, contd.
It’s just that the first derivative (slope) being zero and the second derivative
being negative is the usual way in which the slope can be decreasing through
zero, which is the usual way for there to be a local maximum,
and
the first derivative (slope) being zero and the second derivative being positive
is the usual way in which the slope can be increasing through zero, which is
the usual way for there to be a local minimum.
STROUD
Worked examples and exercises are in the text
Programme 9: Differentiation applications 2
Points of inflexion
A point of inflexion can also occur at points other than stationary points. A
point of inflexion is a point where the direction of bending changes – from
a right-hand bend to a left-hand bend or vice versa.
STROUD
Worked examples and exercises are in the text
Programme 9: Differentiation applications 2
Points of inflexion
At a point of inflexion the second
derivative is zero. However, the
converse is not necessarily true
because the second derivative can
be zero at points other than
stationary points: see right hand of
diagram.
STROUD
Worked examples and exercises are in the text
Programme 9: Differentiation applications 2
Points of inflexion
The test is the behaviour of the second derivative as we move through the
point. If, at a point P on a curve:
d2y
0
2
dx
and the sign of the second derivative changes as x increases from values to
the left of P to values to the right of P, the point is a point of inflexion.
(Added by JAB:) The usual way for this to happen is for the third
derivative to be non-zero.
(Added by JAB:) And all we’re saying overall is that a point of inflexion
is a local maximum or minimum of the SLOPE, i.e. of the first derivative
of y.
STROUD
Worked examples and exercises are in the text
Programme 9: Differentiation applications 2
Differentiation of inverse trigonometric functions
Derivatives of inverse hyperbolic functions
Second derivatives
Maximum and minimum values
Points of inflexion
STROUD
Worked examples and exercises are in the text
Programme 9: Differentiation applications 2
Differentiation of inverse trigonometric functions
(see pp.335/6 for such functions)
NB: sin-1 x is also written arcsin x,
If y  sin 1 x then x  sin y and so
Then:
dy
1

dx dx / dy
1

cos y
1

1  sin 2 y

STROUD
tan-1 x as arctan x
dx
 cos y
dy
d
1
1
sin
x



dx
1  x2
1
1  x2
Worked examples and exercises are in the text
etc.
Programme 9: Differentiation applications 2
Differentiation of inverse trigonometric functions
Similarly:
d
1
cos 1 x 

dx
1  x2
d
1
tan 1 x 

dx
1  x2
STROUD
Worked examples and exercises are in the text