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Theorem on the nth derivative of an k by k determinant

In differential equations, it is useful to be able to find the derivative of a determinant of functions; an interesting exercise is to "find an aestetically pleasing representation of the second derivative of a two by two determinant. " The following theorem is a generalization, being the nth derivative of an k by k determinant. This theorem was formulated by several students and colleagues in connection with my differential equations class in Spring 1993.

Theorem

Examples Illustrating This Theorem