I would like a book that explains how to take the derivative of a function that maps vectors to vectors. Specifically, I would like a book that explains multi-variable differentiation, the multi-variable product rule, and the multi-variable integration by parts.
Here is a simple example of the type of problem I would like to be able to solve with the information in this book: Let $u : \mathbb{R}^n \to \mathbb{R}^n$ such that $u(x) = x^T x$. Find the derivative of u with respect to $x$.
You'd think this material would be in books entitled 'Multivariable Calculus', but I haven't found a book with that title that explains it.
Any help would be appreciated.