- Calculus, Volume 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability
- Tom M. Apostol
- Second Edition
- 1991
- 978-1-119-49676-2
3.6. $\quad$ Exercises
6. $\quad$ Let $f_1, f_2, g_1, g_2$ be four functions differentiable on an interval $(a, b).$ Define \begin{align*} F(x) &= \begin{vmatrix} f_1(x) & f_2(x) \\ g_1(x) & g_2(x) \end{vmatrix} \end{align*} for each $x$ in $(a, b).$ Prove that \begin{align*} F'(x) &= \begin{vmatrix} f_1'(x) & f_2'(x) \\ g_1(x) & g_2(x) \end{vmatrix} + \begin{vmatrix} f_1(x) & f_2(x) \\ g_1'(x) & g_2'(x) \end{vmatrix}. \end{align*}
Proof. $\quad$ We have \begin{align*} F(x) &= \begin{vmatrix} f_1(x) & f_2(x) \\ g_1(x) & g_2(x) \end{vmatrix} = f_1(x)g_2(x) - f_2(x)g_1(x). \end{align*} Then, by the product rule for derivatives, we have \begin{align*} F'(x) &= [f'_1(x)g_2(x) + f_1(x)g'_2(x)] - [f'_2(x)g_1(x) + f_2(x)g'_1(x)] \\ &= [f'_1(x)g_2(x) - f'_2(x)g_1(x)] + [f_1(x)g'_2(x) - f_2(x)g'_1(x)] \\ &= \begin{vmatrix} f_1'(x) & f_2'(x) \\ g_1(x) & g_2(x) \end{vmatrix} + \begin{vmatrix} f_1(x) & f_2(x) \\ g_1'(x) & g_2'(x) \end{vmatrix}. \quad \blacksquare \end{align*}