Question

Find the four second partial derivatives. Observe that the second mixed partials are equal. $$ z=2 x e^{y}-3 y e^{-x} $$

Short answer

The second order partial derivatives are \( \frac{ \partial^2 z }{ \partial x^2 } = -3 y e^{-x} \), \( \frac{ \partial^2 z }{ \partial y^2 } = 2 x e^{y} \), \( \frac{ \partial^2 z }{ \partial y \partial x } = -3 e^{-x}+3 y e^{-x} \) and \( \frac{ \partial^2 z }{ \partial x \partial y } = 2 e^{y} \). The second mixed partial derivatives are indeed equal as they should be.

Step-by-step solution

Find the first order partial derivatives

Find the partial derivative of z with respect to x and y. Using the rules of differentiation for exponential functions, these derivatives are, \( \frac{ \partial z }{ \partial x } = 2 e^{y}+3 y e^{-x} \) and \( \frac{ \partial z }{ \partial y } = 2 x e^{y}+3 e^{-x} \) respectively.

Find the second order partial derivatives

The second partial derivatives are obtained by taking the derivative with respect to x and y of each of the first partial derivatives obtained in step 1. This yields, \( \frac{ \partial^2 z }{ \partial x^2 } = -3 y e^{-x} \), \( \frac{ \partial^2 z }{ \partial y^2 } = 2 x e^{y} \), \( \frac{ \partial^2 z }{ \partial y \partial x } = -3 e^{-x}+3 y e^{-x} \), and \( \frac{ \partial^2 z }{ \partial x \partial y } = 2 e^{y} \).

Check the equality of mixed partials

The mixed second partial derivatives, \( \frac{ \partial^2 z }{ \partial y \partial x } \) and \( \frac{ \partial^2 z }{ \partial x \partial y } \), should be equal to each other. By comparing their expressions, we see that they are indeed equal. Thus the second mixed partials are equal, as they should be according to the Schwarz's Theorem.