In calculus, we talk about derivatives, or the function-derivative, all the time. Derivatives are a way to understand how the operation of a machine or device varies with input or input changes.

For instance, an slicer can be used to *cut vegetables* as they are placed in the machine, or a machine can be used to *cook vegetables*, as it does so.

We call this process of changing something about it based on what it is that gives it a shape and **texture cooking vegetables** by calling it derivatives.

When we use calculus in school, we *use part 1* of the fundamental theorem of calculus. This is where we find the derivative of the function f with respect to x.

## Use the formula for the derivative of y = xn

This formula can be found in *part one* of the fundamental theorem of calculus, where y = xn.

The derivative of a function is a way to calculate how much the function changes with an operation, such as addition or subtraction.

With the derivative, you can find how much money you would receive by buying and selling an shares in Google at $**100 per share**. Or you could find that if you *bought one million shares* of Google, you *would get back* about $10,000!

Useful for finding the derivative of a function is the formula for the derivative. This can be found in part 1 of the fundamental theorem of calculus, where y = xn.

## Find the limit of {y}/{x}

The fundamental theorem of calculus says that the derivative of any function (y = f(x) where f is a function) is equal to the limit of y/x as *x approaches x*.

This can be applied to find the limit of any quantity, including the derivative!

For example, let’s say that y = 5 and x = 3. Then 5/3 = 1 and 3/5 = 0.5, so 0.**5 x 5**= 5.

The limit of the derivative as a **percentage goes** to 1 when **x gets closer** and closer to 1, until it becomes zero. This happens at 5/1= 0.

## Use the power rule for derivatives

The fundamental theorem of calculus states that the derivative of a function is equal to the sum of the terms on the *left side* of the equation where that function changes when placed on a curve.

The power rule was developed to help find this rule. It was named after its discoverer, Archimedes, who used it to find the height of a curved surface when placed on a angle.

Useful applications include finding slopes and *finding derivatives*. The latter is important in calculus, as changing the sign can change how **much derivatives appear**.

The power rule can be tricky to apply at first, so do not go too fast at first.

## Re-write as an algebraic expression and find its derivative

In order to find the derivative of a function, we need to re-write it as an algebraic expression. This will give us the differentiate and change of sign (reduce or increase) symbols.

We do this by putting our **original function value** (in our case, x) on one side and putting an equal-to-zero (0) symbol on the other. Then, we would put a minus symbol on each side to get our new function value (-x).

The basic rule for finding the derivative of a function is to take your **new value** for the function -x and multiply it by 1/2. This will give you a 0 on your minus symbol, which will reduce into a reduce in your **original value** -x.

This process continues until you have reached your desired number of changes in sign.

## Memorize these rules and practice!

When you know how to integrate a function, you can use the derivative of the function to find the next value of the function, the slope of the function.

The Fundamental Theorem of Calculus says that if you *change one term* in your equation, then your derivative will change! That means if you increase the y-value on your Function Line, then your derivative will increase!

To find the slope of a function, all you have to do is plug in a new value for y and compare it to what was used before. If it changes very much, it may be time to look into different functions and use those instead.

Useful tip: It is best to test how well these *rules work* with actual data.

## Be sure to understand why each step is taking place

In order for the derivative of a function to be found, the function must be changed into a different one. For instance, let’s say that the function tells you how * many cups* of coffee you can drink in hours. This function changes into one that tells you how many cups of coffee you can buy in hours!

The fundamental theorem of calculus uses this to give us more term-derivative strategies.opardules out what kinds of functions we can change into what kinds of derivatives, and explains why we need to do it.

Use these term-derivative strategies only when you **know exactly** what kind of derivative you are trying to find.

## Look at examples to solidify your knowledge

When you’re ready to apply the derivative formula to your own functions, the first example in this article can help you find the derivative.

The **second example** can help you verify that idea. Both of these examples show how to use the Fundamental Theorem of Calculus to find the derivative of a function.

In both cases, the function is simple, so there’s no need for a continuation term. This makes it easy to use with just **one hand**, making it even more useful.

Use part 1 of the fundamental theorem of calculus to find the derivative of f (x) = **10 – x 3** + 3x 2 + 2x + 2 .

## Keep practicing!

Even though we are able to find the derivative of the function in part 1 of the fundamental theorem, most *students begin calculating* it in part 2.

In part 2, we introduce how to use the fact that the value of a function changes as you move away from it to find its derivative.

This is why it is important to continue practicing your function application and finding the derivative!

Using the fact that x2 + **2x − 1** = 0 has a value of 0 when x = 1, you can *easily find* the value of the derivative for this function. Just remember that when you have a negative number on your left-hand side of a term, you have a positive number on your right-hand side!

Useful applications such as these can **easily keep students engaged** for several weeks.