For vector-valued functions \(\bm{r}\) and \(\bm{\rho},\) scalar \(c,\) and scalar-valued function \(f,\)
\(\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\Bigl(\bm{r}(t) + \bm{\rho}(t)\Bigr) = \bm{r}'(t) + \bm{\rho}'(t) \)
\(\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\Bigl(c\bm{r}(t)\Bigr) = c\bm{r}'(t)\)
\(\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\Bigl(f(t)\bm{r}(t)\Bigr) = f'(t)\bm{r}(t) + f(t)\bm{r}'(t)\)
\(\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\Bigl(\bm{r}(t) \cdot \bm{\rho}(t)\Bigr) = \bm{r}'(t)\cdot\bm{\rho}(t) + \bm{r}(t)\cdot\bm{\rho}'(t) \)
\(\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\Bigl(\bm{r}(t) \times \bm{\rho}(t)\Bigr) = \bm{r}'(t)\times\bm{\rho}(t) + \bm{r}(t)\times\bm{\rho}'(t) \)
\(\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\Bigl(\bm{r}\bigl(f(t)\bigr)\Bigr) = \bm{r}'\bigl(f(t)\bigr)f'(t) \)