We start with:
#2y^2 - 3x^2y + x siny= 1/(x-y)#
We use the chain rule to implicitly differentiate:
#4yy^(') - 6xy - 3x^2y^(')+ siny + xy^(')cosy = -(1)/(x-y)^2 + (y^('))/(x-y)^2#
Combine terms with y':
#(4y - 3x^2+ xcosy)y^(') +siny-6xy= (y^(') - 1)/(x-y)^2#
Get rid of fraction on RHS:
#(x-y)^2(4y-3x^2+xcosy)y^(') + (x-y)^2siny-6xy(x-y)^2 = y^(') - 1#
Move all y' terms to RHS:
#(x-y)^2siny-6xy(x-y)^2+1=y^(') - (x-y)^2(4y-3x^2+xcosy)y^(')#
Factor out y':
#(x-y)^2siny-6xy(x-y)^2+1=y^(')(1-(x-y)^2(4y-3x^2+xcosy))#
Solve for y':
#y^(')=((x-y)^2siny-6xy(x-y)^2+1)/((1-(x-y)^2(4y-3x^2+xcosy)))#
Simplify:
#y^(')=(siny-6xy+(1)/(x-y)^2)/(((1)/(x-y)^2-(4y-3x^2+xcosy)))#
Solution:
#y^(')=(siny-6xy+(1)/(x-y)^2)/(((1)/(x-y)^2-4y+3x^2-xcosy))#
