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