If is a linear map plus some constant and is times differentiable, prove that:

So I started by induction. Using the chain rule,

Terrific. So next I assumed that the statement was true. Proceeding for , I looked at:

Here's where I run into trouble though. I would like to use the induction hypothesis on the stuff inside the big parenthesis, but the fact that there are now components is throwing me off. Another thing that's not helping is the fact that I'm not really comfortable with this notation. I understand that is the th derivative of evaluated at the point , and I'm applying it to but is a vector, or are each vectors on their own?

So that's where I stand on this problem. Any help is appreciated.