Let

Compute all first-order and second-order partial derivatives. Verify that the mixed partial derivatives are equal.

The first-order partial derivatives are given by

The second partial derivatives are given by

From this we see that the mixed partials are equal:

There is a typo in the answer for df2/dx2 where it should be 4x^2*y^6 instead of 4x^3*y^6. Also for df2/dy2 it should be 9*x^4*y^4 instead of 6*x^4*y^4 (the book is correct too).