The D(f) calling sequence applies the differential operator to the expression f.  The command extends the differential operator D to operate on Vectors and Vector-valued procedures.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{VectorCalculus}\right]\right)\:$

$\mathrm{D}\left(\sin \right)$

$\cos$

$\mathrm{D}\left(⟨x^2\,y^2\,z^2⟩\right)$

$\left(2\mathrm{D}\left(x\right)x\right)e_x\+\left(2\mathrm{D}\left(y\right)y\right)e_y\+\left(2\mathrm{D}\left(z\right)z\right)e_z$

$\mathrm{D}\left[2,1\right]\left(\left(x\,y\right)\mapsto ⟨y\cdot x^2+y^2+3\cdot y\cdot x\,x^2\cdot y^3\,y\cdot x⟩\right)\left(x\,y\right)$

$\left(2x+3\right)e_x\+\left(6x{y}^2\right)e_y\+\left(1\right)e_z$