you may map the nodes on a core (size 2x2) element. Derive the shape functions. then derive the derivatives in the original coordinate via reverse map.
The answer to your question is, well known and main text, in finite element books and notes. Therefore I recommend to read and understand, concepts in finite element method in general and the serendipity elements in specific first. Then you are able to answer not only this and similar questions, but also more paramount questions in the field of numerical analyses.