Gradient Descent Step for word2vec negative sampling
Looks good to me. This derivative is also presented in the paper (equations 56-58).

The paper you're linking to is the most advanced attempt - at least to best of my knowledge - to explain how word2vec works, but there is also a lot of other resources on the topic (just search for word2vec on arxiv.org). If you're interested in word2vec, you may find GloVe interesting too (see: Linking GloVe with word2vec).