La \(\mathbf{u}\) og \(\mathbf{v}\) være deriverbare vektorfunksjoner, og la \(\lambda\) være en deriverbar skalar funksjon. Da gjelder:
\(\frac{d}{dt}\Big(\mathbf{u}(t)+\mathbf{v}(t)\Big) = \mathbf{u}^\prime (t) + \mathbf{v}^\prime (t)\)
\(\frac{d}{dt}\Big(\lambda(t)\mathbf{u}(t)\Big) = \lambda^\prime (t) \mathbf{u}(t) + \lambda(t)\mathbf{u}^\prime (t)\)
\(\frac{d}{dt}\Big(\mathbf{u}(t)\cdot\mathbf{v}(t)\Big) = \mathbf{u}^\prime (t)\cdot \mathbf{v}(t) + \mathbf{u}(t)\cdot \mathbf{v}^\prime (t) \)
\(\frac{d}{dt}\Big(\mathbf{u}(t)\times \mathbf{v}(t) \Big) = \mathbf{u}^\prime (t)\times \mathbf{v}(t) +\mathbf{u} (t)\times \mathbf{v}^\prime (t) \)
\(\frac{d}{dt}\Big(\mathbf{u}\left(\lambda(t)\right)\Big) = \lambda^\prime (t) \mathbf{u}^\prime \left(\lambda(t)\right)\)
Videre har vi at hvis \(\mathbf{u}(t)\neq \mathbf{0}\), så er
\[\frac{d}{dt} \left| \mathbf{u}(t) \right| = \frac{\mathbf{u}(t) \cdot \mathbf{u}'(t)}{|\mathbf{u}(t)|} .\]