Calculus Class Bonus Question

My teacher gave us this bonus question (said getting help was fine as long as we understand the solution), but I think I'm stumped.

"Let f(x) = 1 / sqrt(x)

Determine a function of n for the nth derivative of this function."

I wrote out the first 5 derivatives to find a pattern. Here's the function I've come up with so far:

f(x) to the nth derivative = (-1)^n * ( ) * ( 1/ sqrt(x)^ (2n+1))

I know the (1)^n takes care of the alternating signs, and the ( 1/ sqrt(x)^ (2n+1)) takes care of the power of x. I just can't think of a function to describe the middle term which should be the multiplication of the previous coefficient with the coefficient(?) of the derivative.

Some help would be greatly appreciated :) Thanks!