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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?

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    $\begingroup$ That fraction/slope is a function of $x$. $\endgroup$ – Randall May 30 '19 at 2:07
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    $\begingroup$ Your function $m$ doesn’t tell us the slope of anything directly, it’s just a way of approximating the slope between $x_1$ and $x_2$. (Also it doesn’t need to take in $y_1$ and $y_2$ since those are just $f(x_1)$ and $f(x_2)$ already determined by the function you’re finding the slope of). The limit in the normal definition of the derivative just says “what value do we approach as we take better and better approximations of the slope at a given point $x$?”. $\endgroup$ – Jack Crawford May 30 '19 at 2:12
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    $\begingroup$ @Randall How would I express the fraction/slope as a function? Would it be $g(x) = \frac{f(x) - f(a)}{x - a}$? $\endgroup$ – Calculemus May 30 '19 at 2:13
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    $\begingroup$ Yes, that is exactly the correct expression for the slope of the line through $(a,f(a))$ and $(x,f(x))$. The derivative $f'(a)$ is the limit of all those slopes as $x \to a$. $\endgroup$ – Randall May 30 '19 at 2:13
  • $\begingroup$ Thank you, it's starting to make sense now $\endgroup$ – Calculemus May 30 '19 at 2:16
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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 $(x,f(x))$.

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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.

enter image description here

Sources:

Wiki-Slope Definition

Image Source-Video

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