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# section 13.3 - Assume x represents labor and y capital 3 7...

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Section 13.3 Partial Derivatives

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To find you consider y constant and differentiate with respect to x. Similarly, to find you hold x constant and differentiate with respect to y. Examples: y f x f 2 2 2 2 7 3 ) , ( y x xy z y x y x f + = + - =

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Geometrically speaking, the partial derivatives of a function of two variables represent the slopes of the surfaces in the x- and y- directions.
No matter how many variables are involved, partial derivatives can be thought of as rates of change. Example: Consider the Cobb-Douglas production function When x=1000 and y=500, find a. The marginal productivity of labor b. The marginal productivity of capital.

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Unformatted text preview: Assume x represents labor, and y capital. 3 . 7 . 200 ) , ( y x y x f = As we can do with functions of a single variable, it is possible to take second, third, and higher partial derivatives Notation: xy yy xx f x y f x f y f y f y f y f x f x f x = ∂ ∂ ∂ = ∂ ∂ ∂ ∂ = ∂ ∂ = ∂ ∂ ∂ ∂ = ∂ ∂ = ∂ ∂ ∂ ∂ 2 2 2 2 2 Examples: 1) Find the first four second partial derivatives. x y ye xe y x f--= 3 2 ) , (...
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section 13.3 - Assume x represents labor and y capital 3 7...

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