Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/228

 This may be easily extended to higher derivatives. For instance, since (58) is true,

Similarly for functions of three or more variables.

Given $$u = x^3y -3x^2y^3;$$ verify $$\tfrac{\partial^2 u}{\partial y \partial x} = \tfrac{\partial^2 u}{\partial x \partial y}.$$


 * Solution.
 * $$\frac{\partial u}{\partial x} = 3x^2y -6xy^3, \frac{\partial^2 u}{\partial y \partial x} = 3x^2 - 18xy^2,$$
 * $$\frac{\partial u}{\partial y} = x^3 -9x^2y^2, \frac{\partial^2 u}{\partial x \partial y} = 3x^2 - 18xy^2,$$
 * hence verified.
 * }
 * $$\frac{\partial u}{\partial y} = x^3 -9x^2y^2, \frac{\partial^2 u}{\partial x \partial y} = 3x^2 - 18xy^2,$$
 * hence verified.
 * }