I am looking for a book that explains the equations and identities of the conic section with derivations and explanations of them. I am seeking something similar to the book by Joseph Edwards: Differential calculus for beginners.
I like that book because it had a lot of formulas that are not commonly taught and explained everything with derivations for the most part. Thanks in advance.
As a starting point I'd recommend you browse through archive.org, trying searches for phrases such as "conic sections". The majority of books you'll find were written more than a century ago, but that's fine because that was the hay day of topics like projective geometry and conic sections and the formulae and derivations you're looking for were well known. The Edwards text you mentioned is of the same vintage.
A semi-random sampling:
Smith's Conic Sections, 1904
Runkle's Plane Analytic Geometry, 1888
Heading's Mathematical Methods in Science & Engineering, 1970
Ward's Young Mathematician's Guide, 1771 (250 years old!)
Salmon's Conic Sections,1900
If this isn't what you're looking for, there's much more. And you can try to "tune" your searches with other terms. One stop shopping, and free.