I am looking at some fluid mechanics lecture notes. In this context, $X,Y,Z$ are Lagrangian variables (at time $t = 0$) and $x,y,z$ are Eulerian variables (at time $t > 0$). Euler's identity says that $$\frac{\mathrm{D}J}{\mathrm{D}t}=J\mathbf{\nabla}\cdot \mathbf{u}$$ where $\mathbf{u}=\mathbf{u}(x(t),y(t),z(t),t)$ is the velocity of the fluid, $J=\frac{\partial (x,y,z)}{\partial(X,Y,Z)}$ is the Jacobian determinant and $\frac{\mathrm{D}}{\mathrm{D}t}=\frac{\partial }{\partial t}+\mathbf{u \cdot \nabla}$ is the convective differential operator.
The notes say this:
Whilst this seems to look quite straightforward, I am not sure what property has been used here or why it is permissible for us to be able to differentiate each row of the determinant separately and add up the resulting determinants. Could somebody please offer an input or key result that is being used?
Consider the case $n=2$. Observe \begin{align} \frac{d}{dt} \begin{vmatrix} a_{11}(t) & a_{12}(t)\\ a_{21}(t) & a_{22}(t) \end{vmatrix} =&\ \frac{d}{dt} \sum_{\sigma \in S_2}(-1)^{\operatorname{sgn}\sigma} a_{1\sigma(1)}(t) a_{2\sigma(2)}(t)\\ =&\ \sum_{\sigma \in S_2}(-1)^{\operatorname{sgn}\sigma} [a_{1\sigma(1)}'(t) a_{2\sigma(2)}(t)+a_{1\sigma(1)}(t) a_{2\sigma(2)}'(t)]\\ =&\ \sum_{\sigma \in S_2}(-1)^{\operatorname{sgn}\sigma} a_{1\sigma(1)}'(t) a_{2\sigma(2)}(t)+\sum_{\sigma \in S_2}(-1)^{\operatorname{sgn}\sigma} a_{1\sigma(1)}(t) a_{2\sigma(2)}'(t)\\ =&\ \begin{vmatrix} a_{11}'(t) & a_{12}'(t)\\ a_{21}(t) & a_{22}(t) \end{vmatrix} + \begin{vmatrix} a_{11}(t) & a_{12}(t)\\ a_{21}'(t) & a_{22}'(t) \end{vmatrix}. \end{align} Note I have used the definition: If $A=(a_{ij})$ is an $n\times n$ matrix then \begin{align} \det A = \sum_{\sigma \in S_n} (-1)^{\operatorname{sgn}\sigma} a_{1 \sigma(1)}\cdots a_{n\sigma(n)} \end{align} where $S_n$ is the permutation group of $n$ objects.