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Translating Verbal Descriptions Into Equations With Derivatives

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  • 0:00 Fundamental Point of a…
  • 1:20 From Derivative to…
  • 2:11 Example
  • 3:19 From Verbal…
  • 4:26 Lesson Summary
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Lesson Transcript
Instructor: Kevin Newton

Kevin has edited encyclopedias, taught history, and has an MA in Islamic law/finance. He has since founded his own financial advice firm, Newton Analytical.

If you're new to calculus, chances are you're probably wondering what derivatives are useful for. You've heard that they are rates of change, but maybe you want to know what that looks like in words. Luckily, this lesson can help.

Fundamental Point of a Derivative

While it may seem big and scary at first, all of calculus was created to solve two very basic questions: What is our rate this very instant? and How far have we gone at the rates we've been going at? In a perfectly linear world, we wouldn't need calculus. Of course, things would not work out so well for us in a perfectly linear world. Our speed in our cars would never change, we'd never make more than a certain amount of money, and our bodies would probably collapse from an inability to change rates of the production of certain substances.

In short, we should be happy that we've got calculus. Specifically, we are glad to have derivatives, which let us take a look at the rate of change in a given moment. But wait, all those things I just described don't look like the notation you're used to seeing, do they? Instead, you get d of y over d of x, f prime of x, or y prime. If you need to review how to find a derivative in the first place, you may want to check out some other lessons before proceeding.

While the notation can get a little out of hand, we're going to focus on the use of calculus, especially finding rates. Luckily for us, all we have to do is find the derivative.

From Derivative to Verbal Description

So how do we change a derivative into a verbal description? First, figure out if we are seeing a rate that increases the value or decreases the value. Luckily for us, that's easy - most functions will simply have a negative sign if they're decreasing or will be positive if they're increasing. Now, you may not always get the derivative in the problem, but remember, a derivative is simply a measure of the rate of change.

Therefore, our derivative shows the rate at which something is changing, and the sign in front of it shows if it is either a positive change or a negative change. All we have to do is plug in a value of x for the derivative to find out how fast it is going at a specific time. If we plug in a value of x into the original equation, that would show us position.

Example

So let's say that you are watching a rocket take off, and its distance from the launch pad can be best expressed by the function y = x^2, where x is the number of seconds and y is meters. It's a straightforward function, and if you've been following along in calculus class, you know that the derivative of that function is y prime = 2x. Is that a positive change or a negative change?

If you guessed positive change, you'd be right! This is because this derivative is positive and increasing. So, you've got that function, how do you describe it? For the original y = x^2, we would simply say that the equation shows the position of the rocket for the time, x. But, what if you wanted to know the speed?

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