We need to recognise that this is an implicit function. We can simplify it first:

`3(ln\ x+ln\ y)+sin\ y=x^2`

Taking derivatives:

`3(1/x+1/y(dy)/(dx))+cos\ y (dy)/(dx)=2x`

Collecting terms gives us:

`3/y(dy)/(dx)+cos\ y(dy)/(dx)=2x-3/x`

`(dy)/(dx)(3/y+cos\ y)=2x-3/x`

So

`(dy)/(dx)=(2x-3/x)/(3/y+cos\ y)`

`=(2x^2y-3y)/(3x+xy\ cos\ y)`