First of all, you should specify the/a type of derivative. 'Derivative' does not take an article. The choice for 'type' depends on whether the possible types to be chosen from have been discussed before in the context.
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Well, that's confusing. My study book is R. Murphy's English Grammar In Use, and it explicitly says "you cannot use singular countable nouns alone (without a/the/my etc.)". Maybe it's because the book is about British English?
The 2nd noun just doesn't take an article
Please supply the preceding text that "First of all, you should speci