In the attached image IK means field characteristic zero, [...] means polynomial and IK[x,y] means polynomial in x and y in IK and $m(x,y)$ is generally set to $0$.
If you need more detail or to see in what this is in particular reference it was taken from page 118 in the book ' Concrete Tetrahedron' by Manuel Kauers . Peter Paule.
I derive the equation for Dy which really doesn't make sense or have any meaning or use as far as I can see for one reason how is it valid to set $Dx=1$ or does the $:=$ mean something special ? And factor ring ? I understand what is a factor group in finite groups and in some continuous or infinite groups as long as they are abelian or from normal subgroups but in a ring not sure. Anyway assuming his $D_x,D_y$ means partial derivate put $dm(x,y)=0$ since assuming $m(x,y)=0$ so $$ (\partial m/\partial y)Dy+ (\partial m/\partial y)Dx=0,\; so \;\frac{Dy}{Dx}=-\frac {\partial m/\partial x}{\partial m/\partial y}.$$ Now use the given $Dx=1$ and get his $Dy=-\frac{\partial m/\partial x}{\partial m/\partial y}$ except he has $:=$ in place of just = in most places. Is my derivation correct or is it just coincidence but still can't see the significance or use of this in factor ring or algebraic extension. Extension in this context of what and what is meaning, use and validity of all this even though I probably can understand the meaning of 'factor ring' that alone doesn't explain the meaning, use or validity of all this. Can anyone enlighten or explain all this.