find-the-derivatives-of-the-given-functions-yfrac2-cos-4-x1cot-3-x-3

Question

Find the derivatives of the given functions.
$$y=\frac{2 \cos 4 x}{1+\cot 3 x}$$

Tyler Moulton · Accepted Answer

we want to find the derivative Dy dx with a function Y equals to co sign for X over one plus CO two engine three X two. So if we&#x27;re going to rely on the derivative shortcuts, we picked up through a single variable calculus and particularly going to use our recently learned trigonometry derivatives to solve. So let&#x27;s note the kinetic derivatives and other shortcuts. These rules will sit here. I have also extra derivatives here in particular. We need to use DDX, cossacks and D. D. Expo tangent X. We also need to use the chain will and product rule. Specifically if we rewrite Y as to coast four X times one plus CO two into three X men first, we can start this derivative with the product rule and proceed so, using the product rule on co sign for X and one plus CO two into three X negative. First we obtain derivative. Dy dx equals negative to sign for X times four times one plus tangent X negative first my s to co sign for X times one plus tangent three X. A negative second times negative. Costa can&#x27;t square three X times three. Thus combining like denominators, we have final solution to the bottom two times negative. Four signed for x minus four. Signed for extended three X plus three. Co sign for ex coast. You can square three X all in parenthesis, in the numerator. All over one plus tangent three X square

Todd Vawdrey · Answer

Okay, We&#x27;re going to calculate the derivative of the square root of the CO sign of X over the coastline of four X. You can see that there is the quotient sitting inside of a square root. So we&#x27;re gonna first have to use the chain rule. So the derivative of the square roots is not effect line. Ah, the derivative of the square root is one over two times the square root. So we&#x27;re just gonna leave the inside the same and we&#x27;ll deal with that a little bit later, and then we&#x27;re gonna multiply that by this quotient rule that we get. So we&#x27;re gonna take the derivative of the numerator times the denominator minus the numerator. Don&#x27;t times the derivative of the denominator signed for X right here. And then that&#x27;s all going to be over the denominator squared. So it&#x27;s gonna be over. Cho sang of four x squared right here. Okay, so let&#x27;s do a little bit of simplification right here. Um, so first thing I&#x27;m gonna do is I&#x27;m gonna invert a square root right here on the left hand side. So that&#x27;s going to be, um, 1/2 times the square root of co sign to the forex over co sign of X And then I&#x27;m gonna multiply that by, um, one over the co sign squared of forex right here. And then we&#x27;re gonna times that by the numerator. Um, now, in this ah, we&#x27;re gonna get a negative sign. Ex co sign for X Plus four sign for ex co sign X. Now the only other simplification weaken do deals with the denominator having co sign for X squared and the numerator right here having since it&#x27;s in a square root. Um, weak energy represent that as a co signed to the 1/2 of forex. So if we do just a little bit of simplification on that end, we could actually make this thing as 1/2 of this denominator of the coast sign right here could be the square root of the sea camps of X. And then we&#x27;re going to get, um, a si camp to the, um, three halves power of for ex here. And then we still get that numerator we had from the quotient rule. There are lots of different ways of writing this, um, but they&#x27;re all gonna be equivalent to this version.

Sid Wan · Answer

Yeah, Number twenty four we have. Why? Because square root coz I full x over cose I, max huh? We can right this us co sign for eggs twenty one have because I add studio. I have rights and I think we just need to use when it is quotient Lawrence and he&#x27;s a potion. Co sign tax to the one half square. So it&#x27;s co sign for his cause. I axe. And so for we have Yeah, this guy derivative one have co sign for us to the ninety one half. Multiply derogative off this guys elective side for eggs like before. Just like dollars here or there. We need to channel attack derivative off this guy, which is this court. And you have to motor clouds of boredom bird coz I ax to the one half. Now it&#x27;s a tract. What a tough part remain the same But but of our directive. But I&#x27;m part of the route. It is going to be one half kasai ats to the nephew want have marked by collective sign axe off this huge. How come somebody finds this guy? This&#x27;s too like to x co sign for Max to z you can put two square root t to multiply. Oh, no, Back to Ah, too much. Okay. I&#x27;m just going to leave you here, not Teo off. Negative sign for X coastline X one of the so truck consigning for you are off. What if I want? Because I like to think you are plus my ex, cause I x you can chance those two square root for this one half power. But most importantly, you need to use Chan ruler, like for those bars and Kocian rule in the beginning, okay?

Robert Daugherty · Answer

Section three. That six problem number 29 were daily with two first derivatives where you must use the chain role. So in this case, as everything in this math or capital is, it builds. So not only will I need to use the chain rule, I&#x27;m also going to usually use the product role to get some with an integrated. So why prime going to be equal to Let&#x27;s take that 1st 1? The product rules as take the derivative of that first portion two X times a signed Fourth of Eggs plus X squared. Now what&#x27;s the derivative of the sign to the fourth? That&#x27;s gonna be four times the sign cubed of X Times, the derivative of the sign of X, which is just co sign backs. OK, that takes care of the 1st 2 terms. OK, so that&#x27;s two X times signed fourth of X plus X squared times for signed Cube Co sign. Now let&#x27;s go to the next term. The derivatives of Ellicott that the derivative of X is just going to be one times co sign minus two of X plus. Then you&#x27;re gonna have X the river of co sign to the minus two is going to be minus two co signed to the minus three of X and then the derivative of the co sign is just minus sign of X. Make that a little bit smaller. Size fits, so it&#x27;s gonna be minus sign backs. Now, here again, we&#x27;re in the situation where calculus is all done. It&#x27;s just trig identities and algebraic simplification to see if we can get this looking any better. So why prime is going to be what two X signed, Fourth of X Um, plus four X squared sign cubed of X co sign of X and then it looks like I&#x27;m going to have a plus co sign minus two of acts. And then it looks like, um, plus to sign of x co sign minus three of X. Now, if you think about what you can do is I could say, OK, well, look at the 1st 2 terms. So why prime? I can factor out a two x sign cubed of axe, and then I&#x27;m left with sign of X plus two x co sign of X. And then if I look at these last terms, um, co sign to the negative too. That is the seeking squared of X Plus two. I can write this as the sign of X over the co sign of X and then times the secret squared of X. So now you see, you can factor out, um, a seeking squared of X. So when you do that, your final answer is going to be why prime is two X sign cubed of X sign of X plus two x co sign of X plus seeking squared of X and then you&#x27;re left with one plus two tangent of X. So here&#x27;s a case where, you know, you might argue it, but the the algebraic manipulation of these trick functions was more involved in the actual taking of the first derivative. Okay, so this is my final answer. You realize any of these would have been the right answer? Eso any of these is the right answer, but it&#x27;s just getting it down to the the final form that is the simplest

Sid Wan · Answer

you&#x27;re number fourteen. We have. Why equals Ko Fine for you to the two X. What? This is Looks hard. But you just need to use power rule, cause I some of those you got a negative sign off that part and just take a directive. That part which is forward applied to Tito and we&#x27;re gone. So, like Dave, sign for you two to x eight to the tourists.

Shumayal N · Answer

So we&#x27;re gonna be using the channel for this question. And the outer function is the power 1/5. Still, first we take the derivative of theater function which will give us 1/5 into you&#x27;re right. The inter function as it is, and then using the power will you get minus 4/5 year because 1/5 minus one is minus for over five. And now you take the derivative of the inter function so X to the power for the derivative. This for X cubed, plus you use the chain rule again for co signed to eggs. So first, take the derivative of co sign, which is minus line two eggs and multiply it with the derivative of the inter function, which is to eggs. And that is just too. So that&#x27;s your final answer. But if you wanted to rewrite, it can do so by taking the minus 4/5 power to the gym nominator to make it positive. So it&#x27;ll become positive for five and this entire thing is multiplied with four x cubed minus to sign to next

Problem

Find the derivatives of the given functions. $$y=…

Problem 25 Easy Difficulty

Find the derivatives of the given functions.
$$y=\frac{2 \cos 4 x}{1+\cot 3 x}$$

Answer

$\frac{d y}{d x}=\frac{-8 \sin 4 x(1+\cot 3 x)+6 \cos 4 x \csc ^{2} 3 x}{(1+\cot 3 x)^{2}}$

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Grace H.

Anna Marie V.

Heather Z.

Kayleah T.

Precalculus Review - Intro

Functions on the Real Line - Overview

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$\frac{d y}{d x}=\frac{-8 \sin 4 x(1+\cot 3 x)+6 \cos 4 x \csc ^{2} 3 x}{(1+\cot 3 x)^{2}}$

Basic Technical Mathematics with Calculus

Grace H.

Anna Marie V.

Heather Z.

Kayleah T.

Precalculus Review - Intro

Functions on the Real Line - Overview

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

Grace H.

Anna Marie V.

Heather Z.

Kayleah T.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus