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Find $\frac{d}{d x}\left(\frac{x^{5}-2 x^{2}+3}{4 x}\right)$ by: (a) writing it as a sum of powers and;(b) using the quotient rule.

$\frac{4 x^{5}-2 x^{2}-3}{4 x^{2}}$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 5

Derivative Rules 2

Derivatives

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.

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.

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Okay, So, uh, the D D. X is just telling you to take the derivative. I don't really want to write it all in here. I guess I could. It's not a big deal. I just don't want to rewrite it over and over again. Actually, the fifth minus two x squared plus three all over four x. And the reason why I say I don't want to write it over and over again. This anytime you write equals you're supposed to rewrite it until you finally take the directive, which is not that big a deal. But anyway, the first thing is they wanted you. Thio divide every piece by that for X, which is true in math. So we're looking at 1/4 x to the fourth, because five sorry x to the fifth, divided by x X to the fourth to force reduced the one half x again. I'm doing this metal piece now and then plus 3/4 x to the negative first. So now when I take this derivative, it makes perfect sense that you bring that four in front or looking at X to the third minus one half on. Bring that negative one in front. So we're looking at negative three force because as we multiply extra the native second. Now, if we were to rewrite this a little bit or another equals, you know, it just looked like this X cubed minus one half minus 3/4 x squared eso. This is an acceptable answer. Otherwise, what they want you thio do is the quotient rule what you could do right away, Andi, I would be the directive of the top minus four X. You leave the bottom alone minus the derivative of the bottom, leaving the top alone thanks to the fifth max two x squared plus three all over the denominator squared Notice something square both the four and the 16. And what you'll notice is that you could simplify this. For instance, we could distribute this switch over to green, distribute this in here and distribute this in here. And what you'll notice is that we would have What's that 20 x of the fifth? Yeah, minus 16 X squared minus for extra fifth plus eight X squared, minus 12. And if we divide everything by 16 Well, first of all, I'd have to combine my terms. Uh, so we would have 60 next to the fifth when I divide that by 60. Next squared, I would get just one x cubed, and then I would have eight X squared. Uh, sorry. Negative. Eight X squared, divided by 16 exporters. Negative one half and then negative 12. Divided by 60 X Squared equals this. So these two things are equal once you combine them together. Um, so there you go. This was part B. By the way, this was partake.

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