How to create a big jacobian matrix function over some variables of a vector function?

Question

Suppose I have a vector function of $n$ components and it has $n+m$ arguments,

$$\vec{f}(x_1,x_2,\dots,x_n, y_1, y_2, \dots, y_{m})$$

where $n, m $ are around 10. I want to create a Jacobian matrix over $\vec{x}$ only, $$J_{ij} = \frac{\partial f_i}{\partial x_j},$$ and then use it as a function. However, since I have a lots of arguments, it is really messy easy to make mistake when I create the table. For example, $n=10$ and $m=5$, then $J_{6,5}$ is

J[x1_,x2_,x3_,x4_,x5_,x6_,x7_,x8_,x9_,x10_,y1_,y2_,y3_,y4_,y5_][[6,5]]:=Derivative[0,0,0,0,1,0,0,0,0,0,0,0,0,0,0][f][x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,y1,y2,y3,y4,y5][[6]]

which is a nightmare for me. How to do it in a more efficient way such that I don't have to deal with all the arguments?

The operation D[{f1,f2,...},{{x1,x2,...}}] does not work, like,

u1[x1_, x2_] := x1*x2; 
u2[x1_, x2_] := x1 - x2; 
J[x1_, x2_] := D[{u1[x1, x2], u2[x1, x2]}, {{x1, x2}}]; 
J[1, 1] 

it returns error

General::ivar: 1 is not a valid variable. >>