Applied Mathematics for Business and Economics

Lecture Note Differentiation

Example 1

Suppose that yu u=+ andux=+^317. Use the Chain Rule to find

dy

dx

, then

evaluate

dy

dx

a n is denoted by

x 2

dy

dx

t x=2 (Such an evaluatio

=

)

Solution

yu u uu=+ =+½ and ux=+^317
1
1112
2

+u 1 a

2

dy

du u

−

= =+ nd^32

du

x

dx

=

So, by the Chain Rule,

131 2

2

dy dy du

x

dx du

==

dx u

⎛⎞

⎜⎟+

⎝⎠

and
Ifx= 2 , thenu=+ 217253 = 32 12^2
du
dx

= ×=. Hence

2

1 6

12

⎞

⎟×=

16

1121

x^225105

dy

dx =

⎛⎞⎛

=+ ×=+⎜⎟⎜

⎝⎠⎝⎠

Example 2

Find

dy

when 1

dx

x= if

1

u

y= and ux 32

u+

= − 1 (Answer: 2/3)

en e function

Example 3
Differ tiate th

()( )
4 3
a.fx()=+x^232 x+ b.f xxx=− 2 c. ()
()

5

1

23

fx

x

=

+

Example 4
An environmental study of a certain suburban community suggests that the average

daily level of carbon monoxide in the air will be Cp()= 0.5p^2 + 17 parts per million

when the population is p thousand. It is estimated that t years from now, the

population of the community will be p() 3.1t=+0.1^2

At what rate will the carbon monoxide level be changing with respect to tim
from now?

t thousand.

e 3 years

is to find

Solution
dC
dt

The goal whent= 3. Since

(^) ()() ()
11
(^11221722) 2. 0.5 17
22
ppp
dp

0.5 0.5

dC

p

− −

=+⎡⎣⎤⎦= +

and

0.2

dp

t

dt

=

it follows from the chain rule that

()()

1

(^22)
2

10 .1

0.5 17 0.2

0.5 17

dc dc dp pt

p t

p

−

+ =

2

p

dt dp dt

=×=

+