Defining the derivative

Before I discuss it I should point out one confusion that has happened to me. Every textbook discusses the problem of finding a tangent line before defining the derivative of a function. If you ever watched a video on that, maybe the music video "I'll derive", you should have witnessed a tangent line behaving like a roller-coaster, riding over the graph of a function. Careful there! The tangent line is one thing. The derivative of a function is not the tangent line! When we calculate a limit it yields two possible results: a number or infinity. The definition of a derivative is a limit, but in this case the result of it is another function. It can happen that the derivative yields a number, in which case it's a constant function.