Given: #xsin(y) + cos (xy) = x^2#
Differentiate all of the terms with respect to x:
#(d(xsin(y)))/dx + (d(cos (xy)))/dx = (d(x^2))/dx" [1]"#
For the first term, we must use the product rule,
#(d(gh))/dx = (dg)/dx(h)+(g)(dh)/dx#,
where #g = x#, #h=sin(y)#, #(dg)/dx = 1#, and #(dh)/dx = cos(y)dy/dx#:
#(d(xsin(y)))/dx = (1)sin(y)+(x)cos(y)dy/dx#
Substitute the above into equation [1]:
#sin(y)+(x)cos(y)dy/dx + (d(cos (xy)))/dx = (d(x^2))/dx" [1.1]"#
For the second term, we must use the chain rule:
#(d(cos (xy)))/dx = -sin(xy)(d(xy))/dx#
Then the product rule:
#(d(cos (xy)))/dx = -sin(xy)(y+xdy/dx)#
Substitute the above equation into equation [1.1]:
#sin(y)+(x)cos(y)dy/dx - sin(xy)(y+xdy/dx) = (d(x^2))/dx" [1.1]"#
Use the power rule for the last term:
#sin(y)+(x)cos(y)dy/dx - sin(xy)(y+xdy/dx) = 2x#
Move all of the terms that do not contain #dy/dx# to the right side:
#xcos(y)dy/dx - xsin(xy)dy/dx = 2x-sin(y)+ysin(xy)#
Factor out #dy/dx# from the left side:
#(xcos(y) - xsin(xy))dy/dx = 2x-sin(y)+ysin(xy)#
Divide both sides by the leading coefficient:
#dy/dx = (2x-sin(y)+ysin(xy))/(xcos(y) - xsin(xy))#
