Glad to see you made it to the **business calculus differentiation rules** section. In this section, we will explore the concept of a derivative, the different differentiation rules and sample problems.

In simple terms, a **derivative is a measure of how a function is changing**. In other words, a derivative is a numerical value that says what the **rate of change of a function is for a given input**.

Just when you thought it couldn't get any weirder: the action of finding a derivative is called differentiation. In other words, when you hear the phrase "differentiate", it means that it's time to find a derivative.

First, let's understand what the typical notation of a function is:

$$y=f(x)$$

Remember that in a function, we plug a number into it by applying a rule, and we get a number as a result. For $y=f(x)$, $x$ is a variable that stands for the number we are plugging in (input). It comes in the form $f(x)$ because when we "plug in" $x$, we apply some rule to it. Then, $y$ is the variable that stands for the result we get back (output). Let's do an example:

$$y=f(x)=x+5$$

In the function above, we need to apply the rule $x+5$ to every number we plug in. In other words, the function says "take a number and add $5$ to it", and that gives you your result. For example, let's build a small table with some sample values:

$x$ | $y=f(x)$ | |
---|---|---|

$3$ | $8$ | We plug in $3$, add $5$ to it, and get $8$ as a result |

$20$ | $25$ | We plug in $20$, add $5$ to it, and get $25$ as a result |

$3,000$ | $3,005$ | We plug in $3,000$, add $5$ to it, and get $3,005$ as a result |

As you can see above, we plug in $x$ into the function, the function applies a rule to it (in this case, add $5$ to $x$) and then you receive $y$ as an output. This is why $y=f(x)$.

Knowing this, let's jump into the notations used for derivatives. There are several ways to denote a derivative using math. If $y=f(x)$, then its derivative can be denoted in the following ways:

$$f'(x)$$

$$y'$$

$$\frac{dy}{dx}$$

Let's go in depth into each one:

$f'(x):$ the apostrophe next to the $f$ means that this refers to the derivative of $f(x)$. It is read as "$f$ prime of $x$"

$y':$ the apostrophe next to the $y$ means that this refers to the derivative of $y$ (remember $y=f(x)$, so this is the same as the above). It is read as "$y$ prime"

$\frac{dy}{dx}:$ without getting into too much detail about where this notation comes from, this just means that we are finding the derivative of $y$ in terms of the variable $x$ (it's OK if this does not make too much sense now, as we will see it later on. Just know that it is equal to the two notations above). It is read as "$d$ $y$ over $d$ $x$"

When you see one of the above, you know you are working with a derivative.

In business calculus (and also in economics and social sciences), derivatives have many applications. It can be used to measure:

- How cost and revenue are changing based on how many units are built and sold
- How profit can be maximized for a specific quantity of sales and/or units produced
- How a population is changing over time
- How to distribute area in order to minimize costs when building physical structures

Among many, many other applications, some which we will cover in these tutorials.

In the world of math, you will never really learn anything unless you do it over and over, which makes it second nature at some point. This is especially true when learning differentiation rules.

For that reason, get out some pencil and paper so you can practice the rules as you go. You will not learn them by just reading them, but by actually practicing them.

Let's get started!

- Derivatives are used to measure how a function is changing
- The typical way to denote a function is $y=f(x)$
- Derivatives can have many notations, including $f'(x)$, $y'$ and $\frac{dy}{dx}$
- Derivatives have many applications in business, economics and the social sciences

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