Question: How do I compute the Jacobian

in Maple of a function with respect to a vector that contains product of variables?

I appologize before hand from my abuse of notation. 

Suppose I have a function

f:= x*y*z + 3*x^2*y*z;

and a vector defined as 

v:= <x*z | x^2*z>; #Column vector

and I would like to comput df/dv

So that the result from the Jacobian would be

J=<y , 3*y>; #Row vector

This is a simple example that can be solved by looking, but I would like to know if there is a way in maple to solve something like this. Some of the problems that I find are
1) The variables x*y*z commute i.e. x*y*z=z*x*y=y*z*x=x*z*y
2) And if I apply partial derivatives like  df/(dx*dz) = y+6*x*y and df/(dx*dx*dz) = 6 which is not the result that I desire, because what I want is the partial derivatives with respect to the functions x*z and x^2*z not to the variables x and z.

My application is far bigger than this example that I am posting, this is why I would like to find a way to produce this type of Jacobian. Any suggestion will be highly appreciate it.