Through a natural extension of Clairaut’s theorem we know we can do these partial derivatives in any order we wish to.

In this case the $y$ derivatives of the second term will become unpleasant at some point given that we have four of them. However, the second term has an ${x^2}$and there are three $x$ derivatives we’ll need to do eventually. Therefore, the second term will differentiate to zero with the third $x$ derivative. So, let’s make heavy use of Clairaut’s to do the three $x$ derivatives first prior to any of the $y$ derivatives so we won’t need to deal with the “messy” $y$ derivatives with the second term.

Here is the first derivative we need to take.

Note that if we’d done a couple of $y$ derivatives first the second would have been a product rule and because we did the $x$ derivative first we won’t need to every work about the “messy” $u$ derivatives of the second term.