  By treating generators of the reflection equation algebra corresponding to a
Hecke symmetry as quantum analogs of vector fields, we exhibit the
corresponding Leibniz rule via the so-called quantum doubles. The role of the
function algebra in such a double is attributed to another copy of the
reflection equation algebra. We consider two types of quantum doubles: these
giving rise to the quantum analogs of left vector fields acting on the function
algebra and those giving rise to quantum analogs of the adjoint vector fields
acting on the same algebra. Also, we introduce quantum partial derivatives in
the generators of the reflection equation algebra and then at the limit
$q\rightarrow 1$ we get quantum partial derivatives on the enveloping algebra
$U(gl_N)$ as well as on a certain its extension.
