One suggestion I have (I think it may help but not sure) is to use the multi-variable definition of a derivative. Takes a look at Spivaks Calculus on Manifolds for the definition.
I'm having an incredibly hard time seeing how to get started on this and finding resources elsewhere to get me started. Can anyone at least just give me a nudge in the right direction? I'd appreciate it a ton! I've been staring at it for a few days now.