I assumed that this would be a straightforward proof, as I could just make the substitution l=j and m=l, but upon doing this, I end up with:
δjj δkl - δjl δkj
= δkl - δlk
Clearly I did not take the right approach in this proof and have no clue as to how to proceed.
Starting with LHS:
êi εijk Aj (∇xA)k
êi εijk εlmk Aj (d/dxl) Am
(δil δjm - δim δjl) Aj (d/dxl) Am êi
δil δjm Aj (d/dxl) Am êi - δim δjl Aj (d/dxl) Am êi
Aj (d/dxi) Aj êi - Aj (d/dxj) Ai êi
At this point, the LHS should equal the RHS in the problem statement, but I have no clue where...