9'2 Partial derivatives

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9.2 Partial derivatives
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For a function in one variable f(x) there is no
confusion when finding the derivative (rate of
change) because x is the only variable, so the
derivative is found with respect to that
variable.
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When we find the derivative of f(x,y) the case is
not so clear because we have 2 variables and can
find the derivative with respect to either of
them. These derivatives are called the first
partial derivatives and are denoted a variety of
ways.
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The first partial derivative of z f(x,y) with
respect to x at the point (a,b) is denoted by
or
or more commonly
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The first partial derivative of z f(x,y) with
respect to y at the point (a,b) is denoted by
or
or more commonly
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Before proceeding, recall three basic rules of
differentiation.
c a constant
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When finding either first partial derivative,
treat the other variable as a constant and
differentiate the function using the basic rules
for functions in one variable. If f(x,y) 2x
3y then
and
Why?
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Find the partial derivatives
for the function
What is the rate of change of f in the
x-direction at the point (1,2)? What about the
y-direction?
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Evaluate the first partial derivatives at the
given point.
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Find the values of x and y such that both
and
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Second-order Partial Derivatives.Just as we can
find the second derivative of y f(x), we can
find 2nd order partial derivatives of z
f(x,y). Since 2 variables occur and gives 2 first
partials, we can find two second partials for
each giving a total of four 2nd partials.
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f(x,y)
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f
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Find all four 2nd partials of
Find
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Suppose that revenue from the sale of x model A10
speakers and y model A20 speakers is given in
dollars byDetermine the rate at which the
revenue will change with respect to the number of
A10speakers sold, when 50 A10 and 40 A20 speakers
have been sold.
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The revenue from the sale of x units of a
tranquilizer and y units of an antibiotic is
given bySuppose 9 units of tranquilizer and 5
units of antibiotic are sold.
What is the approximate effect on revenue if 10
units of tranquilizer and 5 units of antibiotic
are sold?
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What is the approximate effect on revenue if 6
units of antibiotic are sold and the number units
of tranquilizer remain constant?
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Suppose the production function of a company is
given by where x represents units of labor and
y represents units of capital. When 4 units of
labor and 3 units of capital are used, (a) Find
the marginal productivity of labor (b) Find the
marginal productivity of capital