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Find $y^{\prime}$ if $y=\frac{\left(x^{2}-1\right)^{3}}{\left(x^{2}+1\right)^{3}}$.

$$\frac{12 x\left(\left(x^{2}-1\right)^{2}\right.}{\left(x^{2}+1\right)^{4}}$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

Missouri State University

Harvey Mudd College

Baylor University

University of Nottingham

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.

01:02

Find $y^{\prime}$.$$

01:01

01:17

Find $y^{\prime \prime}$.<…

01:10

02:49

Find $y^{\prime}(1)$

00:58

Find $d^{2} y / d x^{2}$.<…

05:12

Find $y^{\prime \prime}$

eso there's a couple different ways to find the drifted up. This problem and what I'm gonna do, uh, is not actually what I would do in class. Um, because what I would do is because both the top and bottom are being cubed. How to write this as a single problem. Single question being cute. But instead, what I'm gonna do is is do the quotient rule because we have the questions. And inside the question, all you have, um, a change role. So let's just jump right into that. So if we were asked to find the directive using the quotient rule, what you do is the derivative of the top first and probably should a color goodness. Well, the derivative of the top. You bring the three in front, and then it's a quantity of X squared, minus one being squared. But then we have to do is take the derivative of the inside, which would be two X, And what you need to do is leave the bottom alone. Now that's X squared, plus one cute. And then the question hold says the subtract. Now you leave the top alone X squared, minus one, cubed and Then you take the derivative of the bottom. You'll do that. That blue you bring the three in front sometimes three X squared plus one is now to the second power times the derivative of X squared plus one just two x All over the question rule says Thio, do the denominator X squared plus one cube squared Well, if you take a power to empower you multiply the excellence of Cube Square and makes it to the six power. Now here you condemn. Definitely simplify this because if you look at me every one of these pieces, like the left of the minus to the right of the minus, they all have an X squared mine plus one. And the denominator has an X squared plus one. And what you can do is you can cancel out two of them because that's how many have right here. So this is now to the first power, and then this one is now to the fourth Power. So then, from there we can simplify just a little bit, not not a whole lot in the numerator. You'd have six x x squared, minus one squared, uh, times X squared plus one And then here you'd have and C three temps to excess. Six X again X squared minus one cube. Uh, all over X squared, plus one to the fourth. And while we're at it, you could simplify some more. Okay, Because you can factor out this X squared minus one squared because there's two of them on you can combine like terms and what it boils down to. If you really, really wanted Teoh is you could simplify this to 12 x. You know, how about this? I'll just mention that you can leave your answer like this, or, you know, if you really wanted to could factor out of six X uh, X squared minus one, uh, squared. And then in your other set of parentheses, you'd have Let's see, I factored out. That and two of those. So it'd be like an X squared minus X squared and then one minus native ones, though, turned that into two. Um, and that looks like it matches the answer key. You know, two times six is 12 x x squared minus one squared all over that. So I would let my students leave my answer like this, but your teacher might expect this answer

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