Chapter 12.2, Problem 55ES

Let $u_1,u_2,u_3,{\upsilon }_1,{\upsilon }_2,{\upsilon }_3,w_1,w_2,\text{and }{w}_3,$ be differentiable functions of t. Use Exercise 54 to show that

$\frac{d}{dt}\left|\begin{array}{ccc}{u}_1& u_2& u_3\\ {\upsilon }_1& {\upsilon }_2& {\upsilon }_3\\ w_1& w_2& w_3\end{array}\right|=\left|\begin{array}{ccc}{u^{\prime }}_1& {u^{\prime }}_2& {u^{\prime }}_3\\ {\upsilon }_1& {\upsilon }_2& {\upsilon }_3\\ w_1& w_2& w_3\end{array}\right|+\left|\begin{array}{ccc}{u}_1& u_2& u_3\\ {{\upsilon }^{\prime }}_1& {{\upsilon }^{\prime }}_2& {{\upsilon }^{\prime }}_3\\ w_1& w_2& w_3\end{array}\right|+\left|\begin{array}{ccc}{u}_1& u_2& u_3\\ {\upsilon }_1& {\upsilon }_2& {\upsilon }_3\\ {w^{\prime }}_1& {w^{\prime }}_2& {w^{\prime }}_3\end{array}\right|$