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Use the first derivative to determine where the given function is increasing and decreasing.$p(x)=(x-1) /(x+1)$

Inc. everywhere except when $x=-1$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 2

The First Derivative Test

Derivatives

Campbell University

Baylor University

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.

04:06

Use the first derivative t…

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01:11

Use the derivative to help…

02:17

01:37

Find the interval of incre…

02:52

01:38

Inspect the graph of the f…

03:51

01:03

Use the Monotonicity Theor…

00:56

01:06

Use the derivative to iden…

03:23

01:08

find the derivative of the function P of X. And to do this, we're going to use quotient rule. So we have we're going to have this the bottom equation. BV The top equation. Bu so we have B plus the derivative of you. Which there's only an X. Here. So the derivative is going to be one minus you times the derivative of B. Which is the same idea. There's only an X. Here. So the drug is can be want all over B squared plus one square. So to simplify this we have to over X plus one squared arch equals the derivative. Now we want to find out where this equals zero at what X values. So to solve it we the top doesn't have any um variables in it. So the bottom expose one where that equals zero, is where the function will be undefined or equals zero. So X equals negative one. Is when the derivative will actually not equal zero but be undefined. Since you have A value over zero you have zero the denominator. So with that we're gonna write a number line. We got zero here Or, Sorry, Where are function is undefined and we're going to pick a value slightly below and slightly above. So let's do zero and negative two. So looking at the function, the derivative function right here by seeing if the Values are positive or negative, we can tell if the function is increasing or decreasing. So at -2. Our equation will be positive since anything square will yield a positive number and at zero it will also be positive. So based on this our graph for our equation is increasing on intervals except oh intervals except zero Or sorry, except -1.

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