Programme 7: Differentiation
PROGRAMME 7
DIFFERENTIATION
STROUD
Worked examples and exercises are in the text
Programme 7: Differentiation
Standard derivatives
Functions of a function
Logarithmic differentiation
Implicit functions
Parametric equations
STROUD
Worked examples and exercises are in the text
Programme 7: Differentiation
Standard derivatives
Functions of a function
Logarithmic differentiation
Implicit functions
Parametric equations
STROUD
Worked examples and exercises are in the text
Programme 7: Differentiation
Standard derivatives
As an aide memoir this slide and the following slide contain the list of
standard derivatives which should be familiar.
y  f ( x ) dy / dx
xn
nx n 1
ex
ex
e kx
ke kx
ax
a x .ln a
1
ln x
x
1
log a x
x.ln a
STROUD
Worked examples and exercises are in the text
Programme 7: Differentiation
Standard derivatives
y  f ( x)
dy / dx
sin x
cos x
cos x
 sin x
tan x
sec 2 x
cot x
cosec 2 x
sec x
sec x.tan x
cosecx cosecx.cot x
sinh x
cosh x
cosh x
sinh x
STROUD
Worked examples and exercises are in the text
Programme 7: Differentiation
Standard derivatives
Functions of a function
Logarithmic differentiation
Implicit functions
Parametric equations
STROUD
Worked examples and exercises are in the text
Programme 7: Differentiation
Standard derivatives
Functions of a function
Logarithmic differentiation
Implicit functions
Parametric equations
STROUD
Worked examples and exercises are in the text
Programme 7: Differentiation
Functions of a function
If:
y  f (u ) and u  F ( x)
then
dy dy du

.
dx du dx
For example:
(called the 'chain rule')
y  cos(5 x  4) so y  cos u and u  5 x  4
dy dy du

.
dx du dx
 (  sin u ).5
 5sin(5 x  4)
STROUD
Worked examples and exercises are in the text
Programme 7: Differentiation
Functions of a function
If:
y  ln u and u  F ( x)
then
dy dy du 1 dF

.
 .
dx du dx F dx
For example:
y  ln sin x then
dy
1

.cos x
dx sin x
 cot x
STROUD
Worked examples and exercises are in the text
Programme 7: Differentiation
Functions of a function
Products
Quotients
STROUD
Worked examples and exercises are in the text
Programme 7: Differentiation
Functions of a function
Products
If:
y  uv
then
dy
dv
du
u
v
dx
dx
dx
For example:
y  x 3 .sin 3 x then
dy
 x 3 .3cos3 x  3 x 2 sin 3 x
dx
 3 x 2  x cos3 x  sin 3 x 
STROUD
Worked examples and exercises are in the text
Programme 7: Differentiation
Functions of a function
Quotients
If:
y
u
v
then
du
dv
u
dy
 dx 2 dx
dx
v
sin 3 x
For example:
y
then
x 1
v
dy ( x  1)3cos3 x  sin 3 x.1

dx
( x  1) 2
STROUD
Worked examples and exercises are in the text
Programme 7: Differentiation
Standard derivatives
Functions of a function
Logarithmic differentiation
Implicit functions
Parametric equations
STROUD
Worked examples and exercises are in the text
Programme 7: Differentiation
Standard derivatives
Functions of a function
Logarithmic differentiation
Implicit functions
Parametric equations
STROUD
Worked examples and exercises are in the text
Programme 7: Differentiation
Logarithmic differentiation
Taking logs before differentiating may simplify the working. For example:
x 2 sin x
y
then ln y  ln  x 2   ln(sin x)  ln(cos 2 x)
cos 2 x
d ln y x 2 cos x 2sin 2 x x
1 dy



  cot x  2 tan 2 x  .
dx
2 x sin x
cos 2 x
2
y dx
Therefore:
dy x 2 sin x  x



cot
x

2
tan
2
x


dx cos 2 x  2

STROUD
Worked examples and exercises are in the text
Programme 7: Differentiation
Standard derivatives
Functions of a function
Logarithmic differentiation
Implicit functions
Parametric equations
STROUD
Worked examples and exercises are in the text
Programme 7: Differentiation
Standard derivatives
Functions of a function
Logarithmic differentiation
Implicit functions
Parametric equations
STROUD
Worked examples and exercises are in the text
Programme 7: Differentiation
Implicit functions
There are times when the relationship between x and y is more involved and
y cannot be simply found in terms of x. For example:
xy  sin y  2
In this case a relationship of the form y = f (x) is implied in the given
equation.
STROUD
Worked examples and exercises are in the text
Programme 7: Differentiation
Implicit functions
To differentiate an implicit function remember that y is a function of x.
For example, given:
x 2  y 2  25
Differentiating throughout gives:
2x + 2 y
so.
dy
=0
dx
dy
x
=dx
y
STROUD
Worked examples and exercises are in the text
Programme 7: Differentiation
Standard derivatives
Functions of a function
Logarithmic differentiation
Implicit functions
Parametric equations
STROUD
Worked examples and exercises are in the text
Programme 7: Differentiation
Standard derivatives
Functions of a function
Logarithmic differentiation
Implicit functions
Parametric equations
STROUD
Worked examples and exercises are in the text
Programme 7: Differentiation
Parametric equations
Sometimes the relationship between x and y is given via a third variable
called a parameter. For example, from the two parametric equations
x  sin t and y  cos 2t
the derivative of y with respect to x is found as follows:
dy dy dt
1
 .  2sin 2t.
dx dt dx
cos t
1
 4sin t cos t.
cos t
 4sin t
STROUD
dt
1


since



dx dx / dt 

Worked examples and exercises are in the text
Programme 7: Differentiation
Parametric equations
The second derivative is found as follows:
d 2 y d dy d
 .  . 4sin t 
2
dx
dx dx dx
d
dt
 . 4sin t  .
dt
dx
1
 4cos t.
cos t
 4
STROUD
Worked examples and exercises are in the text
Programme 7: Differentiation
Learning outcomes
Differentiate by using a list of standard derivatives
Apply the chain rule
Apply the product and quotient rules
Perform logarithmic differentiation
Differentiate implicit functions
Differentiate parametric equations
STROUD
Worked examples and exercises are in the text