I occurs to me that, after the first part has be subject to its first derivative, the two terms can be combined and there is no need to do any further derivatives: if an x^8 term is present then x^7 is the answer. If not, then x^6 needs to be investigated.

I take the derivative of the first part after combining (2x-1) * (1-x) to the 4th with (1-x) as a side product.

4(-2x^2 ....)^3(-4x ...)(...-x) +((-2x^2)...)^4(-1)

= [-16*(-8)-16]x^8 ...

for the first part, and for the second:

(16x^4...)(9x^4 ...) = +16*9. So, they do indeed cancel each other out. Onward toward x^6! All we need to prove is that the x^7 in the first part is different from x^7 in the second part. But there are all these bloody cross-products. Oh, sod it -- I've already been to Oxford.