Derivative of cross-product of two vectors

In finding the derivative of the cross product of two vectors $\frac{d}{dt}[\vec{u(t)}\times \vec{v(t)}]$, is it possible to find the cross-product of the two vectors first before differentiating?

You can evaluate this expression in two ways:

• You can find the cross product first, and then differentiate it.
• Or you can use the product rule, which works just fine with the cross product:

$$ \frac{d}{dt}(\mathbf{u} \times \mathbf{v}) = \frac{d\mathbf{u}}{dt} \times \mathbf{v} + \mathbf{u} \times \frac{d\mathbf{v}}{dt} $$

Picking a method depends on the problem at hand. For example, the product rule is used to derive Frenet Serret formulas.

Working from the first principles: \begin{aligned}\vec{u}\left(t+\delta t\right)\times\vec{v}\left(t+\delta t\right)-\vec{u}\left(t\right)\times\vec{v}\left(t\right) & =\vec{u}\left(t+\delta t\right)\times\vec{v}\left(t+\delta t\right)-\vec{u}\left(t\right)\times\vec{v}\left(t+\delta t\right)\\ & \quad + \vec{u}\left(t\right)\times\vec{v}\left(t+\delta t\right)-\vec{u}\left(t\right)\times\vec{v}\left(t\right)\\ & =\left[\vec{u}\left(t+\delta t\right)-\vec{u}\left(t\right)\right]\times\vec{v}\left(t+\delta t\right)\\ & \quad + \vec{u}\left(t\right)\times\left[\vec{v}\left(t+\delta t\right)-\vec{v}\left(t\right)\right] \end{aligned}

Now divide by $\delta t$ and take limit as $\delta t\to 0$, which gives

$$ \frac{d}{dt}\left( \vec{u}\times\vec{v} \right) = \frac{d\vec{u}}{dt}\times\vec{v} + \vec{u}\times\frac{d\vec{v}}{dt} $$

On the other hand $$\frac{d}{dt}\left|\begin{array}{ccc} i & j & k\\ v_{x} & v_{y} & v_{z}\\ u_{x} & u_{y} & u_{z} \end{array}\right|=\left|\begin{array}{ccc} i & j & k\\ \frac{dv_{x}}{dt} & \frac{dv_{y}}{dt} & \frac{dv_{z}}{dt}\\ u_{x} & u_{y} & u_{z} \end{array}\right|+\left|\begin{array}{ccc} i & j & k\\ v_{x} & v_{y} & v_{z}\\ \frac{du_{x}}{dt} & \frac{du_{y}}{dt} & \frac{du_{z}}{dt} \end{array}\right|$$ Using the rule of differentiation of a determinant. One useful application of it is in the proof of Abel's identity (which before Wikipedia was known to me as Ostrogradski-Liouville formula)