From the "I keep having to re-derive this because the Wikipedia explanation sucks" department, here are the transformation rules for differentials and for the gradient in terms of the Jacobian (you can think of this as an adaptation of the simple multivariate chain rule). Of course, the real story is quite a bit more interesting and nuanced, and revolves around the fact that gradient components live in the dual space -- hence the strange inverse transpose Jacobian transformation law. For an elementary, yet rather insightful and detailed discussion I would recommend Koks' Explorations in Mathematical Physics (Chapter 8).
