Functions & the Signs of Their Derivatives

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Read this lesson to learn how you can use the derivative of a function to figure out what the function is doing at any point on the graph. Learn what to look for to determine whether the function is decreasing or increasing.

A Function

When working with functions, one of the many tasks you are asked to do is to figure out whether a certain function is decreasing or increasing within certain intervals. There are several ways you can figure this out. One, you can plug in different values all across the interval to see what the function does. Two, you can graph the function and visually see what the function is doing during the interval. Or three, you can take the derivative of the function and look at the sign of the derivative of the function to find the function's behavior. In this lesson, it's the third way that you'll learn about.

Let's look at this function.


function derivatives


Its Derivative

But first, a definition. The derivative of a function is the slope of a function or its rate of change at any given point. A straight line has a derivative that is constant throughout. But nonlinear functions such as quadratics and rational functions have curves with changing slopes. Taking the derivative of your function will give you the function for its slope. You can then plug in any value into this derivative, and you'll find the slope at that point.

For example, the derivative of your function is this. You found the derivative by following all the rules you already learned for finding derivatives.


function derivatives


To find the slope when the function's x value equals 2, you plug in 2 for x in the derivative function. You then evaluate it. This is what you find.


function derivatives


This tells you that when the function is at the point where x = 2, the slope is 1 / 5.

The Sign of the Derivative

Now, notice the sign of the slope at that point. It's positive. This tells you that your function at that point is rising. 1 / 5 is not a very large slope, so your function isn't going up very fast at that point. You can see this if you graph out the function. At the point x = 2, you can see that your function is just starting to go up. It's not going up very fast, but it is increasing.


function derivatives


A negative slope, or a negative derivative, on the other hand, means the function is decreasing. If you plugged in 0 for x, you'd find that your derivative is negative indicating your function is decreasing at that point. Your graph testifies to that as you can see.

Example

Let's try a problem.

What is the domain in which this function is increasing?


function derivatives


First, you'll need to take the derivative before even trying to answer the question.


function derivatives


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