To answer the question posed in the title, we don’t need to go back to the definition of an oblique asymptote, we just need to know what a function is.
To start learning how to calculate function derivatives (the real ones, not from the definition), you should of course start by preparing a sheet with formulas. What next? I’ll give you a tip.
In this post, I show how to derive the formula for the derivative of the square root of x from the definition of a derivative. Briefly, precisely, and step by step.