We have an implicit function:

sin^-1(x + y) + y = x².

Taking the first term, sin^-1(x + y), and letting

u = x + y,

we differentiate the inverse sine using:

Now du/dx = 1 + dy/dx.

Using the above, and differentiating implicitly term-by-term gives:

Multiplying throughout by:

we have:

Subtracting 1 from both sides:

Grouping the dy/dx terms:

Dividing both sides by:

1 +

We obtain the required solution: