Using the First Derivative to Identify Increasing & Decreasing Functions

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  • 0:01 The First Derivative
  • 2:13 Increasing Function
  • 2:41 Decreasing Function
  • 3:09 Example
  • 3:40 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

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

After watching this video lesson, you will learn how you can find out if the function is decreasing or increasing by just looking at the sign of the first derivative.

The First Derivative

In math, we deal with a lot of functions. Many times, these functions are graphed. For example, the function y = 3x - 4 gives a straight line when graphed.

first derivative

When we start to add exponents to our functions, our graphs begin to look more like roller coasters or some other crazy rides. The function y = -x^5 + 5x^2 - 3 gives you a graph that looks like a crazy roller coaster that makes you feel like you are in free fall!

first derivative

When mathematicians get these functions, and other even more complicated functions, they like to manipulate them and do other mathematical operations on them. One of these operations is taking the first derivative. Other lessons cover taking the first derivative if you need a refresher. The first derivative is the derivative of the original function. Mathematically, the first derivative gives you the slope. What this means for functions with exponents is that the first derivative function tells you what the slope of the function is at every point. As you can see in our last example, our slope changes as we travel along the line, our pretend roller coaster. The first function we saw that graphed into a line has a single slope throughout. This is because our line has the same steepness throughout. It doesn't change. It's like walking up a never-ending hill.

When we call our function f(x), we call our first derivative f '(x). We use one apostrophe to let us know that it is the first derivative. We are not going to go into all the rules for finding the first derivative in this lesson. We have other lessons available for that. Go back and look at them if you need a refresher.

One very interesting thing about the first derivative is that the sign of the first derivative, whether it is positive or negative, actually tells us whether our function is increasing or decreasing. An increasing function has y values that increase as you move towards the right of the graph. A decreasing function has y values that decrease as you move towards the right of the graph. An increasing function has a first derivative that is positive. A decreasing function has a first derivative that is negative.

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