There's a formula for taking the nth derivative of any function of the form y = f(x) g(x) called Leibniz's Rule.
If then , where is some polynomial in u and its derivatives. If you differentiate one more time then you get . So . If you apply this recursive formula to find the first few derivatives then you get
,
,
,
.
That should be enough to convince you that there is not going to be any simple formula for the n-th derivative.