Welcome to MHF. You have a good start.
I wouldn't think of the general derivative in terms of . You start out with and each time you perform another derivative the initial numerator is increased by 2 and the denominator stays the same. For example the first derivative has a power of (-3/2) and a coefficient of (-1/2). The second derivative has a power of (-5/2) and a coefficient of (-1/2)*(-3/2). So how does the power of the derivative relate to the last coefficient? The coefficients follow a pattern and I think the best way to express them generally is using factorials.
Are you familiar with factorials and can you do some more now or are you still stuck?