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 will 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.
The tangent line problem
The definition of a tangent is the rise / run ratio on a right triangle. At school we are given the length of the triangle's sides or we measure it with a ruler. With analytical geometry we know that the distance between two points is [math]\displaystyle{ |a - b| }[/math] in case the line between them is parallel to the axis. When the rise is close to zero, the angle is close to zero. Meaning that a ramp has a very low steepness. The opposite is when the rise's length is so much more than the run that the angle is close to 90°, meaning the highest possible steepness.
[math]\displaystyle{ \text{tan} = \frac{f(x) - f(p)}{x - p} }[/math]
Careful here! The triangle's hypotenuse is not a tangent, it's a secant because it's crossing the graph in two points. Now to make that secant a tangent what we need is a limit to bring the distance between the two points close to zero.
[math]\displaystyle{ \lim_{x \ \to \ p} \frac{f(x) - f(p)}{x - p} }[/math]
What that limit is calculating is the slope of that point. If we could draw a right triangle at a microscopic scale it'd have rise / run ratio equal to that number.
Footnote: about the order of the points. Depending on the textbook they have a graph with concavity up or concavity down. That's why, depending on the textbook, the order of the points in the limit above is reversed. Since the standard notation is [math]\displaystyle{ f(x) }[/math], it's more natural to write [math]\displaystyle{ x \to p }[/math] than the other way around.
The derivative
We can write the same limit as above in a slightly different notation. The notation emphasizes the idea of a limit more than the geometric idea of the rise / run ratio. [math]\displaystyle{ \text{run} = |x - p| }[/math] and [math]\displaystyle{ \text{rise} = |f(x) - f(p)| }[/math]. Let's call run [math]\displaystyle{ \Delta h }[/math]. In physics the letter [math]\displaystyle{ \Delta }[/math] is used to mean a difference between two values, such as two positions [math]\displaystyle{ S_2 - S_1 = \Delta S }[/math]. Now [math]\displaystyle{ x \gt p }[/math] and [math]\displaystyle{ f(x) \gt f(p) }[/math] according to the figure above. With run being called [math]\displaystyle{ \Delta h }[/math], we can write [math]\displaystyle{ f(x) = f(p + \Delta h) }[/math].
[math]\displaystyle{ f'(p) = \lim_{\Delta h \ \to \ 0} \frac{f(p + \Delta h) - f(p)}{\Delta h} }[/math]
The interpretation is that we are making the distance between [math]\displaystyle{ x }[/math] and [math]\displaystyle{ p }[/math] infinitely small but not equal to zero. This is important, when solving exercises with that definition we can do this [math]\displaystyle{ \frac{\Delta h}{\Delta h} = 1 }[/math] because we are not dividing zero by zero.
Footnote: What happened to [math]\displaystyle{ x }[/math] in the above limit? I ended up with [math]\displaystyle{ p }[/math] just because I was referencing the above figure and didn't want to repeat the same graph.
