Can I assume that your "2^n+1" should be "2^(n+1)", not "(2^n)+ 1"?

It would help if you showed HOW you got that. Your answer is clearly incorrect. For example, for n= 1 it gives (2-1)!/2^2 (-a)^1(ax+b)^(-(1+2)/2)= (1/4)(-a)(ax+b)^(-3/2) while the correct first derivative is (1/2)(-a)(ax+b)^(-3/2). For n= 3, your formula gives -(15/2)a^3(ax+ b)^(-7/2)but the answer in the back of the book was

(2 * (2^2n)!) / (((-1)^n) * (a^n) * ((2n)!) * (ax+b)^-(n + 0.5))

Not sure whether ive got it wrong or it's just in a different form... never done nth derivatives before though, so thought I'd check

while the correct third derivative is -(15/8)a^3(ax+b)^{-7/2}.

Interestingly, your formula gives the correctsecondderivative!

(-a)^n= (-1)^n a^n so your formula is only wrong in that (2n-1)!/2^(2n+1) part. (If I assume you really mean (2^n)+ 1, it's worse!)