Applied Mathematics for Business and Economics

Lecture Note Differentiation

()

2

R′ 8 =−×= 25 8 $19.67

3

e. The actual revenue obtained from the sale of the 9th unit is

Δ= − =RR() 9 R( (^8) ) $19.33

4.3 Differentials ............................................................................................

he expression f′(xx)Δ

tion for

T on the right hand

side of the approxima mula

ΔΔf fx x′() i es called the

differential of f and is denoted by df.

Similarly, the expression

ΔΔ i es called the

differential of f and is denoted by df.

Similarly, the expression

s sometims sometim

•

•

y

x

x x+Δx

y=fx()

Δy

Tangent Δx dy

dy

x

dx

Δ on the right-

hand side of the other form of the

approximation formula
dy
yx
dx

ΔΔ

as the differential of y and is denoted by dy.
Thus, is

is known

Δxis small,

Δydywhere

dy

dy=Δx

dx

5 The Chain

uppose the total manufacturing cost at a certain factory is a function of the number

in turn is a function of the number of hours during which the

Rule

S

of units produced, which
factory has been operating. Let C, q and t denote the cost (in dollars), the number of
units, and the number of hours, respectively. Then,
dC
dq

= rate of change with respect to output (dollars per unit)

and
dq
dt

=rate of change of output with respect to time (units per hour)

The pro uct od f these two rates is the rate of change of cost with respect to time. That is,

⎛⎞rate of change of cost

dq dt with respect to time

=⎜⎟

⎝⎠

(dollars per hours)

dC dq

Since the rate of change of cost with respect to time is also given by the derivative

,

dt

it follows that

dC

dC dC dq

dt dq dt

=

This formula is a special case of an important rule called the chain rule.

The Chain Rule ................................................................................................

Suppose y is a differentialbe function of u and u is a differentiable function of

x. Then y can be regarded as a function x and

dy dy du

dx du dx

=