y=2x+cos(xy)y=2x+cos(xy)
Differentiate with respect to x on both the sides.
dy/dx = (d(2x))/dx + (d(cos(xy)))/dxdydx=d(2x)dx+d(cos(xy))dx
dy/dx = 2 dx/dx-sin(xy) * (d(xy))/dx dydx=2dxdx−sin(xy)⋅d(xy)dx Chain rule
dy/dx = 2 - sin(xy) * {x*dy/dx + y dx/dx}dydx=2−sin(xy)⋅{x⋅dydx+ydxdx} Product rule.
dy/dx = 2-sin(xy)*{x dy/dx + y} dydx=2−sin(xy)⋅{xdydx+y}
dy/dx = 2 -x sin(xy) dy/dx - y*sin(xy) dydx=2−xsin(xy)dydx−y⋅sin(xy)
dy/dx + x sin(xy) dy/dx) = 2 - y sin(xy) dydx+xsin(xy)dydx)=2−ysin(xy) collecting dy/dx together on one side of the equation.
(1+x sin(xy)) dy/dx = 2 - y sin(xy) (1+xsin(xy))dydx=2−ysin(xy)
To solve for dy/dx we divide both sides by (1+ x sin(xy))(1+xsin(xy))
We get dy/dx = (2- y sin(xy))/(1+sin(xy)) dydx=2−ysin(xy)1+sin(xy)
