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Find $y^{\prime}$ if $y=\frac{4 x^{3}+5}{2 x^{2}+7}$

$\frac{4 x\left(2 x^{3}-21 x-5\right)}{\left(2 x^{2}+7\right)^{2}}$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 5

Derivative Rules 2

Derivatives

Missouri State University

University of Michigan - Ann Arbor

University of Nottingham

Boston College

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.

00:52

Find $d^{2} y / d x^{2}$.<…

02:28

Find $y^{\prime}$.

01:04

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

01:35

Find $y^{\prime \prime}$.<…

Find $d y$$$y=x^{4}-2 …

02:06

$y=\left(5 x^{2}-7 x+2\rig…

01:23

$$\text {Find } D_{x} y$$ …

02:20

find $y^{\prime}$. $y=\fr…

Find $d^{3} y / d x^{3}$.<…

01:12

03:52

Y=(7x^4-x+3)(-x^5+8)

So we're practicing the quotient rule on this problem because it's a question to the four X cubed plus five and two X squared plus seven. Now, the other piece of this is how far do you go before you're done with the problem? And that's really a teacher preference. Um, so I'm gonna show you a few options, but really, it's up to your professor on where you need to stop eso. It's a quotient telling you to do the quotient rule, which is the derivative of the top, the derivative of a constant zeros. And no need to write that down. Leave the bottom alone, then minus take the derivative of the bottom again, the derivative of a constant zero. So there's no need to write that. And then lay the top alone. We're X cubed plus five on, then all over the denominator square. Now, depending on your professor, uh, you know, you might be able to get by with just leave your answer like this. Otherwise they might say, Oh, no, no, no. You could distribute these and then combine like terms and you could You really can't do anything with the denominator. So just some scratch work here and read. You know, you could see that you get 24 x to the fourth. Plus, let's see, isn't calculated for 12 times seven, which is kind of embarrassing. It's 84 x squared on, then the other piece of B minus 16 x to the fourth minus 20 x eso as you combine like terms we have ah, pair of like terms is eight x to the fourth and you can greatest common factor a four out of all of these and an ex for that matter. So, uh, a better teacher would probably say, Yeah, go ahead and factor out of four X in there and then you'd have to execute less. Let's see, divided by 24 will give me 21 X and then minus five. He's double checking my work here. It looks perfect all over the denominator squared still. So this work here red is for the numerator. Anyway, there's your answer

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