Using definition to find derivative

5. c) If g(x) = 1 / square root (2x+1), use the definition of the derivative to find g prime (1).

I would greatly appreciate a short walk through on this number. I've divided the steps into 5.

Step 1 was to arrange everything so it looks like the formula

g(x+h) - g prime(x) / h

Step 2 was to put everything on common denominator.

Step 3 rationalize - this is where I get stuck

The teacher went over it but just "said the nominator becomes -2h". I tried cracking this and cannot get -2h. There must be something wrong with my algebra, I've tried many many times and by substituing x=1 I still can't.

Step 4 Is to simplify and we should end up with -1 / 3^(3/2).