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Find $y^{\prime}$ if $y=\left(x^{2}-3\right)^{2}\left(x^{2}+2 x+8\right)^{8}$.

$$4\left(x^{2}-3\right)\left(x^{2}+2 x+8\right)^{7}\left(5 x^{3}+6 x^{2}-4 x-12\right)$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

Campbell University

Baylor University

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.

03:12

Find $d y / d x,$ where $\…

02:22

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03:42

01:34

Find $\frac{\mathrm{d} y}{…

01:42

02:05

So this derivative is gonna include the product Will and the general. They're looking at X squared minus three squared times. There's your product X squared plus two X plus eight to the eighth Power. Right. Okay. Yeah. Can I just need toe jump right into this? Okay, so the first thing is, identify your main rules of product rule eso At times, what I'll do is when I ask students to find the derivative, I'll just say, Just write down four blinks, too. And then a plus here in the middle. 34 I don't know if it's really helpful, but, you know, to the left, you have to find that derivative first. So that's a change, Will. Still, because you bring it to in front X squared minus three is now to the first power times. The derivative of the inside is two X and then leave the right side of that problem alone. That x squared plus two X. Let's eat to the eighth power. Um, and then the next piece, you leave the left side alone X squared minus three to the second power, and then you take the derivative of the right side, which is another change rule where you bring eight in front you leave X squared plus two X plus eight alone. It's now to the seventh Power and in times the derivative of the inside, which would be two X plus two. Well, some teachers would actually let you leave your answer like this. They're okay with it. They circle it, move onto the next problem. Other teachers will actually make you look for what's in common between both terms. So they each term shares it two times two. You break down eight as four times to, so there's a four in common. It's the export minus three is shared between each term, so you can write that in front on. Then each term has this X squared plus two X plus eight. But only seven of them are shared. Because this is to the eighth power. This is to the seventh. Um, I'm from there. You just have thio to simplify everything else, which is not necessarily easy. Um mhm. Yeah. What do I have left over? I have Ah. I already took care of two times. Two took care of the experiments. Three I need to distribute this X in here s O, that would be X cubed. Plus two X squared plus eight x then over on this piece. See how I still have X squared minus three. I took care of all of that. I still have a two, and I have this two X plus two. Oh, my goodness is gonna get ugly because I'd have to foil first. That would be to execute plus two x squared, um, see minus six X minus six, but also have to multiply this to buy everything. So we're looking at four x cubed plus four X squared minus 12 X minus 12. That's assuming I haven't made any algebra mistakes, so all of this will be the same. Then I can combine like terms. I hope your teacher doesn't ask you to do this. But as I'm checking with the answer key that says five x cubed, which I didn't get plus six x squared. I didn't get that. I got minus four X, which matches the answer key in minus 12. Like I said, all of this all this part gets moved down as well. I hope your teacher lets you leave your answer like this. Otherwise, um, this is another answer. There you go.

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