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Differentiate.

$ y = \frac {x^2 + 1}{x^3 - 1} $

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$\frac{-x\left(x^{3}+3 x+2\right)}{\left(x^{3}-1\right)^{2}}$

00:31

Frank Lin

01:48

Clarissa Noh

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 2

The Product and Quotient Rules

Derivatives

Differentiation

Saher A.

October 5, 2021

ur probably the worst numerade educator out there, stop reading off a script and actually attempt to teach. I didnt learn anthing

David L.

March 22, 2022

garbage.

you got as far as I got

Missouri State University

Campbell University

Oregon State University

University of Michigan - Ann Arbor

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.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

02:42

Differentiate.$$y=\lef…

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01:13

01:05

Differentiate.$$y=\fra…

Differentiate.$$y=(x-1…

01:11

all right. We want to find the derivative of our function. Why? Why happens to be a quotient? So we have to um make sure remember how to do the quotient rules. Let's review the quotient rule. And basically, and you can use whatever variables you want. But let's say we have uh we want to find the derivative with respect to X. And let's say function you over function B. Well, quote your rule, gives us you prime, V minus B. Prime. You all over B squared. And you might have seen it with Fs and Gs and whatever variables. That's the same process. Okay, So we just have to identify that the top is are you? And the bottom is RV. So let's go ahead and work our quotient rule. So why prime then? Is you prime? Well, we're going to use go ahead and use the power rule. So the derivative of expert plus one, that's just two acts. So that's U times U. Prime. And then V I just write it as is that's the denominator. So we do the derivative of the top times the denominator minus the derivative of the denominator, which is uh three X squared. My power rule times a top all over the bottom squared. And that's certainly a fine answer. If we wanted to clean it up, we could kind of work it a little bit. We'll distribute. So we get two x to the 4th -2 acts. And then I'm going to also distribute here. So I get minus three X. To the fourth minus three x squared to carry that minus through X cubed minus one squared to look for anything interesting cancelations. I guess I can combine a couple of terms. Not a whole lot of interesting stuff, but I can combine those so I get -X to the 4th minus three X squared minus two X. All over X cubed minus one squared. And there's not much else we can do. Maybe a factor and X out of the top and a minus sign. So minus X times X cubed plus three X plus two. That's probably about as far as we can go to clean that up. Okay, fantastic. And hopefully that helped have a wonderful day.

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