Yes Allie, that's exactly what it means. In this case, Implicit differentiation is used because separating the equation would be a pain in the ass.

Instead of applying a regular differentiation and a quotient rule you would have to apply a regular differentiation and the quotient rule inside of the chain rule. Instead of doing a relatively simple differentiation of both sides, you end up having to differentiate y on one side and ((x^2-9)/(x^2+9))^-1 on the other. Ok, it's not really that complex a formula but for pedagogical purposes it illustrates the point well.

It's also worth pointing out that taking the square root of the right hand side before differentiation means the process is no longer implicit, just regular differentiation.