Description
If
x is an indeterminate this is simply the usual differentiation.
i1 : R = QQ[x,y,z];

i2 : diff(x,x^7 + 4*x^3*y  3*y)
6 2
o2 = 7x + 12x y
o2 : R

i3 : diff(x^2+y^2+z^2, y^2*z^2  x^3  1)
2 2
o3 = 2y + 2z  6x
o3 : R

Here is a shortcut that can save some typing.
i4 : diff_x x^6
5
o4 = 6x
o4 : R
