We first find d/dxddx of each term.
d/dx[xy^2]-d/dx[(1-xy)^2]=d/dx[C]ddx[xy2]−ddx[(1−xy)2]=ddx[C]
d/dx[x]y^2+d/dx[y^2]x-2(1-xy)d/dx[1-xy]=0ddx[x]y2+ddx[y2]x−2(1−xy)ddx[1−xy]=0
y^2+d/dx[y^2]x-2(1-xy)(d/dx[1]-d/dx[xy])=0y2+ddx[y2]x−2(1−xy)(ddx[1]−ddx[xy])=0
y^2+d/dx[y^2]x-2(1-xy)(-d/dx[x]y+d/dx[y]x)=0y2+ddx[y2]x−2(1−xy)(−ddx[x]y+ddx[y]x)=0
y^2+d/dx[y^2]x-2(1-xy)(-y+d/dx[y]x)=0y2+ddx[y2]x−2(1−xy)(−y+ddx[y]x)=0
The chain rule tells us:
d/dx=d/dy*dy/dxddx=ddy⋅dydx
y^2+dy/dx d/dy[y^2]x-2(1-xy)(-y+dy/dxd/dy[y]x)=0y2+dydxddy[y2]x−2(1−xy)(−y+dydxddy[y]x)=0
y^2+dy/dx 2yx-2(1-xy)(-y+dy/dx x)=0y2+dydx2yx−2(1−xy)(−y+dydxx)=0
dy/dx 2yx-2(1-x)dy/dx x=-y^2-2y(1-xy)dydx2yx−2(1−x)dydxx=−y2−2y(1−xy)
dy/dx( 2yx-2x(1-x))=-y^2-2y(1-xy)dydx(2yx−2x(1−x))=−y2−2y(1−xy)x
dy/dx=-(y^2+2y(1-xy))/(2yx-2x(1-x))dydx=−y2+2y(1−xy)2yx−2x(1−x)
For (1,-1)(1,−1)
dy/dx=-((-1)^2+2(-1)(1-1(-1)))/(2(1)(-1)-2(1)(1-1))=-1.5dydx=−(−1)2+2(−1)(1−1(−1))2(1)(−1)−2(1)(1−1)=−1.5
