I think it is a typo.

I put fu=fdS.

Stick it into the identity you made up to find a value to curl(fdS).

Re-arrange to find an identity for (gradf)xdS.

Then stick (gradf)xdS into an integral sign, use your second identity and Stokes theorem, and note that curl(dS) = 0 (I think).

Then the solution should pop out.