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.

I'm mentioning that confusion because I think very often some people are mislead, thinking that to derive a function is the same thing as finding the tangent line. Not quite. When we have functions such as polynomials of degree greater than 2 and any transcendental function, the process of calculating a derivative yields another function that is not linear, it's not a straight line! There is no such thing as finding a function that is tangent to another in multiple points.

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{ \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.

Notation: [math]\displaystyle{ \frac{dy}{dx} }[/math] it came from [math]\displaystyle{ y = f(x) }[/math]. It's called the Leibniz's notation for differentiation. If you look at the finding the tangent problem, this notation is clearly connected to the idea of infinitesimal increments. Writing [math]\displaystyle{ \frac{\Delta y}{\Delta x} }[/math] is clearly a ratio or quotient, the tangent. It gives the idea of the variation of something in respect to something else. As we can conclude, the ratio is larger when the numerator is larger. Conversely, the ratio is smaller when the denominator is larger. Graphically, it's the function's steepness. Each function having its own particular steepness, its own rate of change.

It may happen that some people invert the ratio in the same way they invert the tangent's ratio. However, with functions this is harder to do because we always plot y as the vertical axis and x as the horizontal axis. If you understand the concept of a function, it's hard to invert the ratio because it's always [math]\displaystyle{ \frac{df}{dx} }[/math]. Notice that we are never doing a division by zero, because we always consider [math]\displaystyle{ dx \neq 0 }[/math]. In other words, there isn't a rate of change if don't have a variation in x to begin with.