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:

`1/sqrt(1-mu^2)(du)/(dx)`

and `(du)/dx = 1 + dy/dx`.

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

`1/sqrt(1-(x+y)^2)(1+(dy)/(dx))+(dy)/(dx)` `=2x`

Multiplying throughout by:

`sqrt(1-(x+y)^2)`

we have:

`(1+(dy)/(dx))+sqrt(1-(x+y)^2)(dy)/(dx)` `=2xsqrt(1-(x+y)^2`

Subtracting 1 from both sides:

`(dy)/(dx)+sqrt(1-(x+y)^2)(dy)/(dx)` `=2xsqrt(1-(x+y)^2)-1`

Grouping the `dy/dx` terms:

`(1+sqrt(1-(x+y)^2))(dy)/(dx)` `=2xsqrt(1-(x+y)^2)-1`

Dividing both sides by:

`1+sqrt(1-(x+y)^2)`

We obtain the required solution:

`(dy)/(dx)=(2xsqrt(1-(x+y)^2)-1)/(1+sqrt(1-(x+y)^2`

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