This post is meant for readers familiar with algebra and derivatives, but want to deepen their understanding and/or need a refresh.

Linear functions

Let's start with a family of very basic functions: the linear functions, expressed as $f (x) = a x + b$ . You might remember its derivative is $f^{'} (x) = a$ , because $x$ is multiplied by $a$ and the constant $b$ "disappears" when taking the derivative. This is correct, but let's actually calculate the derivative. Since $f (x)$ is a linear function, $f^{'} (x)$ is the same for all $x .$ That is, a linear function "goes up" with the same "speed" everywhere, as can be seen in the following graph for $g (x) = 2 x + 5$ :

For example, between $x = 0$ and $x = 1$ , $f (x)$ increases with $2$ , just like it does between e.g. $x = 2$ and $x = 3$ . Therefore, determining the average slope between $x$ and $x + d$ will do. The average slope between $x$ and $x + d$ is how much $f (x)$ increases between $x$ and $x + d$ , divided by the difference between $x$ and $x + d$ (which is $d$ ). Let $d = 1$ , in which case we don't have to do the division, as $f^{'} (x) = \frac{f (x + 1) - f (x)}{1} = f (x + 1) - f (x)$ . Filling in $a x + b$ for $f (x)$ and $a (x + 1) + b$ for $f (x + 1)$ , we get:

$f^{'} (x) = f (x + 1) - f (x) = (a (x + 1) + b) - (a x + b) = a x + a + b - a x - b = a$

There it is! $f^{'} (x) = a$ . So for $g (x) = 2 x + 5$ , where $a = 2$ , this means $g^{'} (x) = 2$ .

Polynomials (and more)

Polynomials are functions with the following form:

$f (x) = a_{0} x^{n} + a_{1} x^{n - 1} + . . . + a_{n - 1} x + a_{n}$

Determining their derivative is a bit more tricky than determining the derivative of a linear function, because now, the derivative isn't necessarily the same everywhere. After all, take $g (x) = x^{2}$ :

We can see this is a curved line, and so the derivative is constantly changing. We can sill do something like the "trick" we did with linear functions, but we can't determine $f^{'} (x)$ by looking how $f (x)$ changes between $x$ and $x + 1$ : that would assume $f^{'} (x)$ is the same between $x$ and $x + 1$ , which isn't true. For $x$ and $x + 0.001$ , we would have a better estimate of $f^{'} (x)$ , but we'd still assume $f^{'} (x)$ to be constant between these values. We need to determine how $f (x)$ changes between $x$ and some $x + d$ , w...

Calculus in Game and Decision Theory