Prog 7 Matematika Industri 2 | Blog Mas'ud Effendi Prog 7 MEF

DIFFERENTIATION

STROUD

Worked examples and exercises are in the text

Differentiation
Standard derivatives

Functions of a function
Logarithmic differentiation

Implicit functions
Parametric equations

STROUD

Worked examples and exercises are in the text

Differentiation
Standard derivatives

Functions of a function
Logarithmic differentiation

Implicit functions
Parametric equations

STROUD

Worked examples and exercises are in the text

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

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

Differentiation
Standard derivatives

Functions of a function
Logarithmic differentiation

Implicit functions
Parametric equations

STROUD

Worked examples and exercises are in the text

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

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

Differentiation
Functions of a function

Products
Quotients

STROUD

Worked examples and exercises are in the text

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

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:
then
y
x 1
v

dy ( x  1)3cos3 x  sin 3 x.1

( x  1) 2
dx

STROUD

Worked examples and exercises are in the text

Differentiation
Standard derivatives

Functions of a function
Logarithmic differentiation

Implicit functions
Parametric equations

STROUD

Worked examples and exercises are in the text

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

Differentiation
Standard derivatives

Functions of a function
Logarithmic differentiation

Implicit functions
Parametric equations

STROUD

Worked examples and exercises are in the text

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

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

Differentiation
Standard derivatives

Functions of a function
Logarithmic differentiation

Implicit functions
Parametric equations

STROUD

Worked examples and exercises are in the text

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

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dt
1



since


dx dx / dt 


Worked examples and exercises are in the text

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

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