Rewrite the equation so that it is equal to 0:
#cos(xy)-sin(x^(1/3))(sin(y))^(1/2)=0#
Differentiate each term:
#(d(cos(xy)))/dx-(d(sin(x^(1/3))(sin(y))^(1/2)))/dx=0#
For the first term let #u = xy# and use the chain rule:
#(d(cos(u)))/dx = (d(cos(u)))/(du)(du)/dx#
#(d(cos(u)))/(du) = -sin(u) = -sin(xy)#
#(du)/dx = (d(xy))/dx = dx/dxy+xdy/dx = y + xdy/dx#
Returning to the chain rule:
#(d(cos(u)))/dx = -sin(xy)(y + xdy/dx)#
#(d(cos(u)))/dx = -ysin(xy) - xsin(xy)dy/dx#
Substitute this into the equation:
#-ysin(xy) - xsin(xy)dy/dx-(d(sin(x^(1/3))(sin(y))^(1/2)))/dx=0#
Multiply by -1:
#ysin(xy) + xsin(xy)dy/dx+(d(sin(x^(1/3))(sin(y))^(1/2)))/dx=0#
For last term, use the product rule:
#(d(sin(x^(1/3))(sin(y))^(1/2)))/dx = (d(sin(x^(1/3))))/dx(sin(y))^(1/2)+ sin(x^(1/3))(d((sin(y))^(1/2)))/dx#
#(d(sin(x^(1/3))(sin(y))^(1/2)))/dx = (x^(-2/3))/3cos(x^(1/3))(sin(y))^(1/2)+ 1/2sin(x^(1/3))(sin(y))^(-1/2)cos(y)dy/dx#
Substitute this into the equation:
#ysin(xy) + xsin(xy)dy/dx+(x^(-2/3))/3cos(x^(1/3))(sin(y))^(1/2)+ 1/2sin(x^(1/3))(sin(y))^(-1/2)cos(y)dy/dx=0#
Move the terms that do not contain #dy/dx# to the right:
#xsin(xy)dy/dx+ 1/2sin(x^(1/3))(sin(y))^(-1/2)cos(y)dy/dx=-(x^(-2/3))/3cos(x^(1/3))(sin(y))^(1/2)-ysin(xy)#
Factor out #dy/dx# from the terms on the left:
#(xsin(xy)+ 1/2sin(x^(1/3))(sin(y))^(-1/2)cos(y))dy/dx=-(x^(-2/3))/3cos(x^(1/3))(sin(y))^(1/2)-ysin(xy)#
Divide by the leading coefficient:
#dy/dx=-((x^(-2/3))/3cos(x^(1/3))(sin(y))^(1/2)+ysin(xy))/(xsin(xy)+ 1/2sin(x^(1/3))(sin(y))^(-1/2)cos(y))#
