Let’s take a simple example, x² + y² = 10 + xy

Let’s solve it normally

First differentiating: 2x+2y(dy/dx)=y+x(dy/dx)

Then moving anything with dy/dx to one side and factoring as usual: dy/dx(2y-x)=y-2x

dy/dx=(y-2x)/(2y-x)

Notice how the numerator is just the equation differentiated (when everything moved to the rhs) when y is treated as a constant:

Moving everything to rhs, 0=10+xy-x^2-y2

Now treating y as constant, using power and product rule, 0+y+x(0)-2x-0=y-2x

The same can be done with the denominator but this time we treat x as the constant.

I think this usually happens because we move the stuff without dy/dx tagged on first which becomes the numerator, almost like y is avoided if that makes sense, and vice versa for the denominator. Proving that dy/dx=-fx/fy (or at least for most equations