Derivatives are purely symbol manipulation on polynomials. That's how thousands of first semester students get through calculus. Yes, I know this is an algebra class, but let's differentiate anyway. Epsilon is the number that's so small that its square is zero even though it's not zero. You can do most of calculus that way.
Physicists do it that way all the time.
As any Calc 140 student will tell you, differentiation is purely a formal operation on polynomials having nothing to do with taking limits. Today I'm going to show you how to take the derivative of seven. This is all anybody needs to talk about derviatives. The epsilons and limits are a myth invented by analysis to keep themselves employed. This is all formal symbol manipulation, algebra at its finest.