Math 142 Lecture Notes for Section 8.2
Section 8.2 -
1
Partial Derivatives
Definition 8.2.1:
Given z = f (x, y), we define the first-order partial derivatives of f with respect to x
and y in the following way.
Example 8.2.2:
Find the first-order partial derivatives of the following functions.
(a) f (x, y) = x3 − y 3
(b) f (x, y) = 2x2 − 3x2 y + 5y + 1
(c) f (x, y) =
4x2 − y 2
x2 + 2y 2
Math 142 Lecture Notes for Section 8.2
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(d) f (x, y) = ln(3x2 + xy − y 8 )
Example 8.2.3:
The productivity of a video game manufactoring company is given approximately by the
Cobb-Douglas production function
f (x, y) = 25x0.4 y 0.6
(a) Find
∂f
∂f
and
.
∂x
∂y
(b) If the company is currently using 1000 units of labor and 2500 units of capital, find
the marginal productivity of labor (the partial derivative of f with respect to
labor) and the marginal productivity of capital (the partial derivative of f with
respect to capital).
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(c) For the greatest increase in productivity, should the management of the company
encourage increased use of labor or increased used of capital?
Definition 8.2.4:
Given z = f (x, y), there are four second order partial derivatives:
Example 8.2.5:
Find all second-order partial derivatives of the function, f (x, y) =
x4
y3
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It is even possible for a function to have more than one or two independent variables.
For instance, a function of three independent variables could be w = f (x, y, z). Here
w would have three partial derivatives, one for each independent variable, treating the
others as constants.
Example 8.2.6:
Find all of the first-order and second-order partial derivatives for the function w(x, y, z) =
xey + yez .
Suggested Homework Problems: 1-37 (odd), 47