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Then, starting from this he calculates the partial derivative with respect to L and the cross second-order partial derivative (the partial derivative with respect to K), whose results are shown below: $$\frac{\partial^2 V}{L^{2}}=-\frac{\alpha }{bL} X^{-\frac{c}{b}}Y^{\frac{1}{b}-1}\left ( X\frac{dY}{dX}-cY \right )$$

Unlike the case of functions of a single variable, we can also take the second order cross-partial derivative. This is defined as. This tells us how the slope ...

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Schwarz's theorem states that if the second derivatives are continuous, the expression for the cross partial derivative is unaffected by which variable the ...

A further application in economics of partial derivatives is to two-variable functions that represent the relative demands of two commodities that are ...

Young's theorem: Corresponding cross partial derivatives are equal. (To read more about Young’s theorem, see Simon & Blume, Mathematics for Economists, p 330.) Suppose y = f (x1,…,xn) y = f ( x 1, …, x n) is a continuously differentiable function of n n variables.

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If j ≠ i j ≠ i, then xixj x i x j -second order partial derivative is called the cross partial derivatives.

The partial derivative of utility with respect to consumption of good x can be interpreted as the marginal utility of x, or the amount of utility gained when a ...

The volume V of a cone depends on the cone's height h and its radius r according to the formula The partial derivative of V with respect to r is which represents the rate with which a cone's volume changes if its radius is varied and its height is kept constant. The partial derivative with respect to equ…

The last item is called a cross-partial derivative: you differentiate first with x and then with z (or the other way around: you get the same result – Young’s Theorem). 22 2 22 and 0 and ; xx x x x YY Yze e ze zx YY Y ze e zx x z Total Differential. Consider . yfxz (, )

The last item is called a cross-partial derivative: you differentiate first with x and then with z (or the other way around: you get the.

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The interpretation of the cross partial here is the change in the return (or marginal productivity) of labor when capital increases marginally.

Cross partial derivatives ... Just as we can differentiate the derivative of a function (if the derivative is differentiable) to get the second derivative, we can ...

Mple 15.21

Then, starting from this he calculates the partial derivative with respect to L and the cross second-order partial derivative (the partial derivative with respect to K), whose results are shown below: