Definition of derivative as a slope: is a slope a function?

Question

I'm trying to understand the definition of derivative as a slope.

The definition is as follows:

Let $a \in \mathbb{R}$. Let $f$ be a function defined, at least, on an interval centered at $a$. The derivative of $f$ at $a$ is the number:

$$f'(a) = \lim\limits_{x \to a} \frac{f(x) - f(a)}{x -a}$$

When I learned about limits, I learned that they apply to functions. How come it also applies to a slope?

I was thinking that maybe a slope can be interpreted as a function with four inputs, like this:

$$m(x_1, x_2, y_1, y_2) = \frac{y_2 - y_1}{x_2 - x_1}$$

Is that a valid interpretation?

Answer 1

As discussed in the comments, $$ f'(a) = \lim_{x \to a} g(x) $$ where $$ g(x) = \frac{f(x) - f(a)}{x-a}. $$ Note that $g(x)$ is the slope of the line through the points $(a,f(a))$ and $(x,f(x))$.

Answer 2

I was thinking that maybe a slope can be interpreted as a function with four inputs

The equations you put are correct, except that the slope is usually not written as a function of four variables (by convention). In cases, it may require 4 parameters to get its value. The slope is defined as:"the slope or gradient of a line is a number that describes both the direction and the steepness of the line".

The equation of the a line with slope m is:

$$y=mx+b$$

The following picture is important. It is fundamental in learning Calculus. Almost any decent Calculus book will have a fancy illustration of these facts.

Sources:

Wiki-Slope Definition

Image Source-Video