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I have

x =(x^2+y^2)^[1/2]

I differentiate

1= 1/2 (x^2+y^2)[-1/2] (2x+2yy')

So far so good. I try to multiply this out.

1= (2x)/2 (x^2+y^2)[-1/2] + (2yy'/2)(x^2+y^2)[-1/2]

I solve for y'

y'= 1/{(x (x^2+y^2)[-1/2]} / {y(x^2+y^2)[-1/2] }

1/x (x^2+y^2)[1/2] * 1/y (x^2+y^2)[1/2]

The square roots multiply out.

y'=(x^2+y^2)/xy

yet the correct response is

y'=[(x^2+y^2)^1/2 -x] / y

Am I missing something?

Thanks