Applied Mathematics for Business and Economics

Lecture Note Differentiation

2

df dg

df gf

dx dx

dx g g

⎛⎞ −

⎜⎟=

⎝⎠

Example 5
Differentiate the rational function

3

(^2212)
−

−+

=

x

xx

y

Solution
According to the quotient rule,

()()()()

()

()( )()

()

() ()

22

2

2

2

22 2

22

3221221

3

32 2 2 21

3

246 221 615

33

dd

dy xxxxx x

dx dx

dx x

xx xx

x

xx xx xx

xx

−+−−+−

=

−

−+−+−

=

−

−−−−+ −+

==

−−

− 3

3 The Derivative as a Rate of change

3.1 Average and Instantaneous Rate of Change ...........................................

Suppose that y is a function of x, sayy=fx( ). Corresponding to a change from x to

x+Δx, the variable y changes by an amount Δyfx x fx=+Δ−()(). The resulting

average rate of change of y with respect to x is the difference quotient

change in ( ) ( )

Average rate of change

change in

y f xxfx

xx

+Δ −

==

Δ

As the interval over which you are averaging becomes shorter (that is, as Δx
approaches zero), the average rate of change approaches what you would intuitively
call the instantaneous rate of change of y with respect to x and the difference

quotient approaches the derivative ()or

dy

fx

dx

′. That is,

( ) ( )

Instantaneuous rate of change limx 0 ()

fx x fx dy

fx

Δ→ x dx

+Δ −

==′

Δ

=

Instantaneous Rate of Change

Ify=fx(), the instantaneous rate of change of y with respect to x is given by

the derivative of f. That is,

Rate of change ()

dy

fx

dx

==′

Example 1
It is estimated that x months from now, the population of a certain community will be

Px x( )=+ +^220 x8, 000

a. At what rate will the population be changing with respect to time 15 months

from now?

b. By how much will the population actually change during the 16th month?