The problem with your original working is that you differentiated the quotient without using the quotient rule. You can't just differentiate the numerator and the denominator separately. Here's my working:

y^2 = (x^2 - 9) / (x^2 + 9)

First:
let <i>f</i> = x^2 - 9
let <i>g</i> = x^2 + 9
let <i>h</i> = y^2
Observe that with respect to x:
f' = 2x
g' = 2x
h' = 2y(dy/dx)
By the quotient rule:
h' = (gf' - fg')/g^2

Therefore:
2y(dy/dx) = [2x(x^2+9) - 2x(x^2-9)]/(x^2+9)^2

And finally, simplifying:
2y(dy/dx) = 36x / (x^2+9)^2
y(dy/dx) = 36x / 2(x^2+9)^2

dy/dx = 18x / y(x^2+9)^2

Take that with a grain of salt - I haven't done implicit differentiation in several years. What does your book say?