First we take the change in xx of both sides.
d/dx[5]=d/dx[-yx^2]-d/dx[(xy)/(y-1)]+d/dx[y^2x]ddx[5]=ddx[−yx2]−ddx[xyy−1]+ddx[y2x]
0=-yd/dx[x^2]+x^2d/dx[-y]-y/(y-1)d/dx[x]+xd/dx[y/(y-1)]+y^2d/dx[x]+xd/dx[y^2]0=−yddx[x2]+x2ddx[−y]−yy−1ddx[x]+xddx[yy−1]+y2ddx[x]+xddx[y2]
0=-y(2x)+x^2d/dx[-y]-y/(y-1)(1)+xd/dx[y/(y-1)]+y^2(1)+xd/dx[y^2]0=−y(2x)+x2ddx[−y]−yy−1(1)+xddx[yy−1]+y2(1)+xddx[y2]
0=-2xy+x^2d/dx[-y]-y/(y-1)+xd/dx[y/(y-1)]+y^2+xd/dx[y^2]0=−2xy+x2ddx[−y]−yy−1+xddx[yy−1]+y2+xddx[y2]
The chain rule tells us that d/dx=d/dyxx(dy)/(dx)ddx=ddy×dydx. That gives us:
0=-2xy+x^2(dy)/(dx)d/dy[-y]-y/(y-1)+x(dy)/(dx)d/dy[y/(y-1)]+y^2+x(dy)/(dx)d/dy[y^2]0=−2xy+x2dydxddy[−y]−yy−1+xdydxddy[yy−1]+y2+xdydxddy[y2]
0=-2xy+(dy)/(dx)(-x^2)-y/(y-1)+(dy)/(dx)x(((y-1)d/dy(y)-(yd/dy(y-1)))/(y-1)^2)+y^2+(dy)/(dx)(2yx)0=−2xy+dydx(−x2)−yy−1+dydxx⎛⎜⎝(y−1)ddy(y)−(yddy(y−1))(y−1)2⎞⎟⎠+y2+dydx(2yx)
0=-2xy+(dy)/(dx)(-x^2)-y/(y-1)+(dy)/(dx)x(((y-1)(1)-y(1))/(y-1)^2)+y^2+(dy)/(dx)(2yx)0=−2xy+dydx(−x2)−yy−1+dydxx((y−1)(1)−y(1)(y−1)2)+y2+dydx(2yx)
0=-2xy+(dy)/(dx)(-x^2)-y/(y-1)+(dy)/(dx)x((y-1-y)/(y-1)^2)+y^2+(dy)/(dx)(2yx)0=−2xy+dydx(−x2)−yy−1+dydxx(y−1−y(y−1)2)+y2+dydx(2yx)
0=-2xy+(dy)/(dx)(-x^2)-y/(y-1)+(dy)/(dx)x((-1)/(y-1)^2)+y^2+(dy)/(dx)(2yx)0=−2xy+dydx(−x2)−yy−1+dydxx(−1(y−1)2)+y2+dydx(2yx)
0=-2xy+(dy)/(dx)(-x^2)-y/(y-1)+(dy)/(dx)(-x)/(y-1)^2+y^2+(dy)/(dx)(2yx)0=−2xy+dydx(−x2)−yy−1+dydx−x(y−1)2+y2+dydx(2yx)
(dy)/(dx)(-x^2)+(dy)/(dx)(-x)/(y-1)^2+(dy)/(dx)(2yx)=2xy+y/(y-1)-y^2dydx(−x2)+dydx−x(y−1)2+dydx(2yx)=2xy+yy−1−y2
(dy)/(dx)(-x^2-x/(y-1)^2+2yx)=2xy+y/(y-1)-y^2dydx(−x2−x(y−1)2+2yx)=2xy+yy−1−y2
(dy)/(dx)=(2xy+y/(y-1)-y^2)/(-x^2-x/(y-1)^2+2yx)dydx=2xy+yy−1−y2−x2−x(y−1)2+2yx
