TITLE: How can you calculate the derivative of this Wronskian?
QUESTION [5 upvotes]: If $W(y_1,y_2,y_3)=\left| \begin{array}{ccc} y_1 & y_2 & y_3 \\ y_1' & y_2' &  y_3'  \\ y_1'' & y_2'' & y_3'' \end{array}\right|$, how can I show that $W'(y_1,y_2,y_3)=\left| \begin{array}{ccc} y_1 & y_2 & y_3 \\ y_1' & y_2' &  y_3'  \\ y_1''' & y_2''' & y_3''' \end{array}\right|$? I am not sure how to go about computing the derivative of a determinant. Any input would be greatly appreciated.

REPLY [4 votes]: $$W^\prime(y_1,y_2,y_3)=\frac{d}{dx}\left| \begin{array}{ccc} y_1 & y_2 & y_3 \\ y_1\prime & y_2\prime &  y_3\prime  \\ y_1\prime\prime & y_2\prime\prime & y_3\prime\prime \end{array}\right|\\
=\left| \begin{array}{ccc} y_1\prime & y_2\prime & y_3\prime \\ y_1\prime & y_2\prime &  y_3\prime  \\ y_1\prime\prime & y_2\prime\prime & y_3\prime\prime \end{array}\right|+\left| \begin{array}{ccc} y_1 & y_2 & y_3 \\ y_1\prime\prime & y_2\prime\prime &  y_3\prime\prime  \\ y_1\prime\prime & y_2\prime\prime & y_3\prime\prime \end{array}\right|+\left| \begin{array}{ccc} y_1 & y_2 & y_3 \\ y_1\prime & y_2\prime &  y_3\prime  \\ y_1\prime\prime\prime & y_2\prime\prime\prime & y_3\prime\prime\prime \end{array}\right|\\
=\left| \begin{array}{ccc} y_1 & y_2 & y_3 \\ y_1\prime & y_2\prime &  y_3\prime  \\ y_1\prime\prime\prime & y_2\prime\prime\prime & y_3\prime\prime\prime \end{array}\right|.
$$