I used the chain rule first for each of the two functions (g(x) and f(x)), getting
g'(x) = 3/2x^2(x^3 + 2)^-1/2
f'(x) = -3/2x^2(x^3 + 7)^-3/2
Then, using the product rule, I get f(x)g'(x) + f'(x)g(x) =
= [(x^3 +7)^-1/2][3/2x^2(x^3 + 2)^-1/2] + [-3/2x^2(x^3 + 7)^-3/2] [(x^3 + 2)^1/2]
which is just a mess, and I don't know how (or if it's possible) to simplify it.
Is that correct so far?