Let f(x,y)=y3+3xy2-x3=3.
f(x+h,y+k)=3, where the small change h in x creates a small change k in y.
f(x+h,y+k)=(y+3y2k+3yk2+k3)+3(x+h)(y2+2yk+k2)-(x3+3x2h+3xh2+h3)=3.
If h and k are sufficiently small we can ignore as insignificant terms containing h2, k2 and any terms involving the product of h and k.
f(x+h,y+k)=y3+3y2k+3xy2+6xyk+3hy2-x3-3x2h=3.
The differential is f(x+h,y+k)-f(x,y)=0:
3y2k+6xyk+3hy2-3x2h=0.
Divide through by 3h:
y2k/h+2xyk/h+y2-x2=0,
(y2+2xy)(k/h)=x2-y2, and (k/h)=(x2-y2)/(y2+2xy).
Limit as h,k→0 makes k/h=dy/dx=(x2-y2)/(y2+2xy).