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Use the first derivative to determine where the given function is increasing and decreasing.$v(x)=x^{4}+5$

Dec on $x < 0,$ inc on $x > 0$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 2

The First Derivative Test

Derivatives

Missouri State University

Harvey Mudd College

University of Nottingham

Idaho State University

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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Use the first derivative t…

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01:25

Use the derivative to help…

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03:36

This is Chapter three, Section two, problem nine. And in this problem, we are asked to find where this function VFX is increasing in where it is decreasing by finding the first derivative. The first thing that we're going to do is find are critical points, and if you can remember, the critical points are where the derivative of the function is equal to zero. So we'll start by finding are derivative of VFX, which is going to be the prime of X, and that's just going to be equal to four X Cube. The only time when four X cubed is equal to zero is when X is equal to zero. So that's a pretty easy critical point. So we have a critical point at X equals zero. From here, we can draw our number line. So here we'll have X equals zero, and then we can choose a number that is less than zero. So let's just say X equals negative one. And for greater than zero, we can choose X equals one, and we're going to plug in both of those values into V Prime of X to find whether our function is increasing or decreasing in that area. Yeah, so if we take the prime of negative one, we'll have four times negative. One cubed negative one cubed is just negative. One negative. One times four is going to be equal to negative four because that is a negative value. That means that any time we'll have an X less than zero, that means our function is going to be decreasing on the other side. If we plug in one, we get four times one cube. That's just going to be four. This is a positive value, which means that any time we have an X that is greater than zero, our function is going to be increasing. So in conclusion, our function VFX is going to be decreasing anytime. X is less than zero and it's going to be increasing. Okay, anytime X is greater than zero, and that is our answer.

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