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Find $\frac{d}{d x}\left(\frac{4 x^{6}+3 x^{3}-8 x}{6 x^{5}}\right)$ by: (a) writing it as a sum of powers and;(b) using the quotient rule.

$\frac{2 x^{5}-3 x^{2}+16}{3 x^{5}}$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 5

Derivative Rules 2

Derivatives

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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.

03:09

Find $\frac{d}{d x}\left(\…

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Find $d y / d x$ by (a) us…

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Use the quotient rule to s…

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(A) Find $f^{\prime}(x)$ u…

00:14

Simplify using the quotien…

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00:22

06:54

Consider$$g(x)=\left(\…

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03:35

Use the Quotient Rule to e…

02:07

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Compute the given derivati…

00:35

Divide using the quotient …

it's our task is to find the derivative of this question a couple different ways. Eso The first thing is we have four x of the six less three X cubed minus eight events all over six x to the fifth eso. What they're trying to get you to realize is that this is the same thing is the derivative of dividing every piece by that six x to the fit. So what we're looking for four or six reduces to two thirds X. The next one will be one half X to the native second and then 46 produces four thirds exit the negative four. So if you're doing this derivative, don't change colors toe green that direct. It will just be two thirds if you bring that negative two in front half of negative two is one a negative extra. The native third and then same thing is a negative times. Negative. Four times for 16. It's a positive 16 3rd X to the negative fifth. So this is an acceptable answer. You might also see somebody rewrite. This is two thirds minus one over execute plus 16/3 X to the fifth or what you might see is the quotient rule. So if you do the quotient rule, that's taking the derivative of the top. So that will equal 24 x to the SIS plus nine X squared minus eight and multiply by the denominator and a minus. Take the drift of other bottom, which is 30 x of the fourth and leave the top alone Forex the six plus three X cubed minus eight x all over the denominator squared and you just square each piece so it becomes 36 x to the 10th because you square each piece and you multiply the explains. And I'm not gonna go through this. But if you were to distribute all of these pieces distribute, distribute, distribute. And if you really want, you could actually divide everything by, you know, X to the fourth. Um, so that becomes six X and this becomes extra. The six, but you'll notice is ah, you actually get the same thing is what's right here. After you factor out, it just looks a little different. But it is the same answer eso you can trust me. This is a good enough answer. We're done. We can move on

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