What is the derivative of y = sqrt(x^2+2x-1) ? I think the answer key is wrong!

I am doing practice problems out of a book to prepare for the AP Calculus 1 Test. For the problem above I got:

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

My book, however, says the answer is:

y' = (x+1)/y

In the answer key the book always explains how to get the correct answer. In this case it says to rewrite the thing as:

(x^2+2x-1)^(1/2)

and use the formula for derivatives of exponents:

d/dx u^(p/q) = (p/q) * [ u ^ ( [p/q] - 1) ] du/dx

THIS IS EXACTLY WHAT I DID BUT A GOT A COMPLETELY DIFFERENT ANSWER!!!!! :'-(

Do I just suck at life, or is my book wrong, or does my answer somehow equal the book's and I just need to use some wacked algebraic manipulation to get the book's answer?

Any help would be very much appreciated. Thanks in advance :)

PS- So it's easy to find:

My answer: y' = (x+1)/[sqrt(x^2+2x-1)]
The book's answer: y' = (x+1)/y

The equation can be written as y^2 = x^2 + 2x - 1

Now differentiate both sides of this equation with respect to x. This gives

2ydy/dx = 2x + 2

dy/dx = (x + 1)/y

That's the same thing! Notice that y = sqrt(x^2 + 2x - 1), so the book's answer is just a shorthand version of what you got. Good job!

Yeah!