#tan(x-y)=y/(3+x^2)#
Take the derivative of both sides.
#d/dx(tan(x-y))=d/dx(y/(3+x^2))#
Solve.
#d/dx(tan(x-y))=(1-dy/dx)(sec^2(x-y))#
#d/dx(y/(3+x^2))=((3+x^2)(dy/dx)-(y)(2x))/(3+x^2)^2#
Now we know:
#(1-dy/dx)(sec^2(x-y))=((3+x^2)(dy/dx)-(y)(2x))/(3+x^2)^2#
We can simplify
#(1-dy/dx)(sec^2(x-y))(3+x^2)^2=((3+x^2)(dy/dx)-(y)(2x))#
#((sec^2(x-y))(3+x^2)^2-(dy/dx)(sec^2(x-y))(3+x^2)^2)=((3+x^2)(dy/dx)-(y)(2x))#
#((sec^2(x-y))(3+x^2)^2+(y)(2x))=((3+x^2)(dy/dx)+(dy/dx)(sec^2(x-y))(3+x^2)^2)#
#((sec^2(x-y))(3+x^2)^2+(y)(2x))=(dy/dx)((3+x^2)+(sec^2(x-y))(3+x^2)^2)#
#((sec^2(x-y))(3+x^2)^2+(y)(2x))/((3+x^2)+(sec^2(x-y))(3+x^2)^2)=dy/dx#
Depending on how simplified the answer needs to be, this is technically the solution.
