evaluate-the-derivatives-of-the-given-functions-for-the-given-values-of-x-use-the-product-rule-che-7

Question

Evaluate the derivatives of the given functions for the given values of $x$. Use the product rule. Check your results using the derivative evaluation feature of a calculator.
$$S=\frac{2 n^{3}-3 n+8}{2 n-3 n^{4}}, n=-1$$

Gregory Higby · Accepted Answer

All right, you&#x27;re gonna want to have your calculator ready to evaluate D. S. D. N. When N equals negative one. Because if we noticed that s is a giant quotient. Uh and you&#x27;re going to have to cube some things you have to divide. It&#x27;s not the prettiest problem. Let&#x27;s put it that way. three m plus eight All over to n minus three and To the 4th power. So as you notice, I mentioned the quotient in here, we&#x27;re going to do the quotient rule. So uh stick with me. We&#x27;re going to take the derivative of the top which is your power rule multiplied by the Coefficient with the exponent and then subtract one from the exponent. Leave the denominator alone minus Now take the derivative of the bottom. Again, I&#x27;m just following the power rules that we&#x27;ve already learned. Leave the top alone, all over the denominator two and -3 and to the fourth. And that quantity squared. Um so from here I know a lot of people would try to simplify, I wouldn&#x27;t, I would just be plugging in negative one and for all of these ends I know there&#x27;s a lot of them but stick with me. The reason why I mentioned that it&#x27;s not that bad is because a few of the answers are going to be the same. Like negative one squared is just possible one, Six times one is 6 -3 is three. Um So again you have a negative one. So we&#x27;re looking at negative two and negative one to the fourth, power is positive one. So you get negative five right there. But if you notice you actually get negative five the same term this is the same term as in the denominator. So that must also be negative five when you square it. now be careful because -1 when you cube it is a Still a negative. So that will turn that into plus 12. So we&#x27;re looking at 14 because it&#x27;s two plus 12. So here&#x27;s another um -2 and becomes plus three plus eight. Uh So what are we looking at there? Nine? So as I simplify, I&#x27;m seeing negative 15, I don&#x27;t trust my math to do 14 times nine in my head. It&#x27;s -126, All over 25. And this is why I would use a calculator. Uh negative 141, divided by 25, Gives me an answer of negative 5.64. And as I check that it is correct, let&#x27;s circle that and go onto the next problem.

Brian Ketelobeter · Answer

Okay. Good day. Ladies and gentlemen, this is problem number 3 73 Um, and it asked me it asked to find a derivative of the given function there. So, um OK, so first off, um, you know, F is equal to, um the summer two functions we have, um, at F at X eyes equal to G uh, X plus h of X. Um, of course, we have, uh, g of X to be equal to three x cubed g of thanks is equal to three x cubed, pre x cubed. And we have a church of acts is equal to negative obsessed, sir. Negative for X to the it is ticket. Okay, so, um, since this is true, since we have this, um, the some rule for differentiation says that af ah, prime of X is equal to g prime at the acts plus h crime at axe. So our first well, I guess our major task here, um, is to calculate the two corresponding things. You drive well, you there is each crime. So Okay, so what can we do? Well, um well, basically, I mean, really, this is very, you know, pretty simple to do. Um because G is a, um, you know, is a oh, no meal. And so, uh, it&#x27;s a polynomial. Also, we get g prime acts. My no meal and yes, you&#x27;d say is equal to nine X to the second tendentious. Using the the power, the power rule, I guess, Um, it is over. Very. I&#x27;m very tired. Sorry about that. Um, and secondly, h prime. We do the same thing. We do something very similar we use. I guess the more generalized, uh, notion of power rule are the generalized differential power rule. And of course, this one is, um, negative aches. Uh, X to the course. It&#x27;s Nega its end minus one in this case. So it&#x27;s negative. Two minus one. Um, which, of course, is going to equal to negative three. Right. So, um, now, let&#x27;s take these guys, uh and, um, plug them in, both of them back in to get after crime. So, um, so we just plug it in so we get that half prime house crime. It&#x27;s ah is equal to now g prime. Of course, we&#x27;ve already computed, um and then plus the h prime, um, books. Oh, no. I made a mistake. I&#x27;m sorry. Ah, yeah. Dull. I made a mistake there, but I&#x27;m I&#x27;m fixing the answer here. H prime is not negative. Eight. It&#x27;s eight. I&#x27;m I forgot to change the, um uh, the negative Signed to a positive sign. Ha! Again, you failed to recognize my mistake. How air you. Ah, yeah. So f prime, then is just this guy here? Um and that&#x27;s about it. I mean, that&#x27;s really all there is to it, so thank you very much. Have a nice day.

Amrita Bhasin · Answer

we know we&#x27;re gonna have to apply the chain role for this. So taking the derivative we bring the explain it down becomes the new coefficient, make the export, and one less so instead of three now becomes too. And then we multiply by the driven of a four mice X squared, which, as you know, is negative two acts. The derivative of Forrester, Cyrix. It&#x27;s a constant simplifying to f prime of axe equals negative sex acts times four minus X squared, squared.

Jessica Wellington · Answer

Okay, so we have. Why is he going to e race to x squared over X cubed plus two. So for this, we&#x27;re going to start by recalling this rule right here which says that if you have e an exponential to a power so eat of the G, which will denote for whatever&#x27;s in the power of the function, the derivative is G prime. So the derivative of what&#x27;s in the exponents times, the original function eight of the G. So for something like this that has a exponents right here that&#x27;s fairly complicated. It&#x27;s useful to solve this by itself. So if we go ahead and let g be equal two x squared over X cubed plus two, we noticed this is going to be a chain rule. So we&#x27;re first going to find the derivative of this so g prime is going to equal low. So what&#x27;s on the bottom? Hello, D hi. Derivative of the high to X minus. Hi. What&#x27;s on the top? Deal. Oh, three x squared. If you&#x27;re wondering what I&#x27;m saying, um, you can look back at some previous videos for the new line that I use for the chain rule. This is all over the square of what&#x27;s below this square. What&#x27;s below? So now this is what we have as our g prime in this case. So we have all of this times e to the original function. So we&#x27;re going to simplify this a little bit before we put it all together. We&#x27;re going to distribute the two X so we have to x to the fourth plus four X minus three eggs to the fourth over. Ex cute plus two quantity squared A little bit more simplification. If you notice we have this right here in this right here that gives us four eggs minus X to the fourth over X cubed plus two square. Again, this is just one step and finding the derivative of the function. We&#x27;ve only found the derivative of the exponents So on the next page, or to take that and we&#x27;re going to write it all together. So remember that if we have into the gene, the derivative is g prime e to the G. We found g prime already. So now are derivative. Why? Prime is equal to G prime, which is what we found in red, which is four x minus four over X squared. Plus two square. Thanks, Cube. Sure. X cubed plus two square. So this is G Prime times E to the G. Which was this Each the G part is just the original function that we originally wanted to find. The derivative up X cubed plus two. So now we&#x27;ve put it all together here for our final answer.

Linh Vu · Answer

on a function f X n coach aged about off X square over the x about Tripolis June and the question asked, You find the AARP around the X recording. Ever have the able off some function on the ANC&#x27;s The real deal? We go to the agent about the function of the ANC&#x27;s Tam&#x27;s, the UPM off X Here we can identify in this power there will be no function on the banks. Therefore, by formally we get aged about X squared over exploratory rest do times with the X squared off expert typist to derivative. Here we can tell even undecided. And we even write that one x square our expert tributes to derivative and noticed them. We still need to apply that caution ruin this case here and by the caution group which you get a go to expert rivers to live out to. And they lived on the expert Could you two x terms with the expert trippers to minors that the review in the expert trip was to encoded a tree x square time for the expert you Therefore we get equal a Jew. We went inside and we have the two Expo for plus four x minus three Expert for over the expert chippers to square and by simplification we get to go to for ex miners Expert for the writing by expound Childress to about two Now we&#x27;re blood is on back into here. Therefore we ghenda Evram the X we go to a bar on the X square number. Expert trip is two times with the for X minus expert for the invited by expert trippers to our too. And this will be the expression from the Ephraim Axe.

Jeffrey Utley · Answer

this question gives us a function with several terms and asks us to calculate its derivatives. To start off, we have to look at the function as of X is equal to X squared plus two x cubed times co tangent inverse of X minus the natural log of one plus X squared of all the terms here. But luckily the derivative is something that we can do all three derivatives separately and then add them together. So let&#x27;s do them all separately. So what is the derivative of X squared? Well, we moved the two down by the power rule and then we subtract one from the exponents. Tu minus one is just one. So are derivative will be two X. The second term will be a good get about more complicated here since we have to use the power rule. So power rule is if we have new times being then you times the prime for the derivative of U times V is equal to you times the prime plus you prime times the so our you, in this case is to execute RV is inverse co tangent of X. So to execute times what is the derivative of inverse pretension of X. It is negative 1/1 plus X squared. Plus, Now we calculate the derivative of to execute the power rule. We moved the three down, multiplying it to to to get sick. Then we have X to the three minus one. So then we have X squared. Then we just have inverse co tangent of X. So I&#x27;m gonna clean it up a little bit here by moving this negative two x cubed up to the numerator. And so this is our derivative of the longer term. Lastly, we must find the derivative of this last term negative Ellen of one plus X squared. So for this, we&#x27;re gonna have to use a change. First, we just use the normal rule for natural log. We have the negative just outside one over, whatever was in the parentheses. So that&#x27;s one plus X squared. But then we have to multiply by the derivative of the inside of the parentheses. So the derivative of one plus x squared the derivative of 10 selectable go away the derivative of X squared. We have right here, so we multiply it by two x to clean it up a little bit up with the negative two x on top. So now we have all three of these derivatives were looking for. We just have to add them all together. So this will be f Prime of X is equal to two X since that was our first term. Plus, I&#x27;m gonna do printed seasons. We have two terms in the second derivative, this which is negative two x cubed over one plus x squared. And then to that, we add six x squared time inverse co tension of X. Lastly, we&#x27;ll add the last turn, which was this negative two x over one plus X squared. So I will try to simplify this down a little bit, but there&#x27;s not much we can do. However, we have a common denominator one over X squared between these two terms. So to reorganize this, I&#x27;m gonna put those lab. So first we have to x from right here that I&#x27;m gonna add this term plus six x squared co tangent in bursts of X. Now, for the last two. If you notice they both have a minus sign. So a way you can distribute that out is just do minus then two X cubed plus two X over one plus X squared, and then this is going to be our derivative of the function.

Problem

Evaluate the derivatives of the given functions f…

Problem 31 Easy Difficulty

Evaluate the derivatives of the given functions for the given values of $x$. Use the product rule. Check your results using the derivative evaluation feature of a calculator.
$$S=\frac{2 n^{3}-3 n+8}{2 n-3 n^{4}}, n=-1$$

Answer

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

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Kristen K.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

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Basic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus