find-the-derivatives-of-the-given-functions-r04-e2-theta-ln-cos-theta-3

Question

Find the derivatives of the given functions.
$$r=0.4 e^{2 	heta} \ln \cos 	heta$$

Tyler Moulton · Accepted Answer

All right. We want to find dy dx four. Y equals 0.4. Either the two X. Natural algorithm of coastline of X. So we&#x27;re going to rely on derivative shortcuts to solve in particular. Well, zero through three listed here on the left, zero is G D X P T U equals B U l n b d s G exceeded the EU is either the U d X rules one through three Other power world product rule and change respectively. Y equals point for each of the two. XL and Cossacks has already written it&#x27;s most easily differential form. Thus we can proceed to solve, we see that we&#x27;re gonna have to use the product wolf to separate out either the two X and Ellen Cossacks derivatives. So we have dy dx equals point for each of the two X times two Ln Cossacks is using real zero on each of the two X plus point for each of the two X times negative seven X. Over Cossacks. This is using the fact that the derivative of Ln. You is do you over you where coastline has a lot of negative sign that we can simplify this as dy dx equals each of the two X times 20.8 Ln cossacks minus 0.4 tangent of X. Remember that Synnex over Cossacks equals tantric.

Robert Daugherty · Answer

section 3.6 number 45 We&#x27;re dealing with derivative that requires to know the chain role. So we&#x27;re asked to find the derivative our prime, as a function of Fada. So this is gonna be the product rule. You&#x27;ve got two different functions multiplied. So what we want to do first is let&#x27;s take the derivative of the first term. So the derivative of sine squared. So what&#x27;s the derivative sign is the coastline. So this is gonna be the co sign of data squared Now, what is the derivative of Data Square? That&#x27;s to thinner times. The co sign of two things was product rules. So this is you think that this is f and G, and this is F prime G plus G prime F. So plus the derivative of the second term. So the derivative of the co sign is minus sign of tooth Ada, the derivative of Tooth Ada is going to be too times the sign of data squared and the trouble just a little bit of organization that can go on here. So final answer r squared of Fada is going to be equal to toothache. The co sign of they the squared the co sign of tooth data and then minus two sign of data squared sign of two data and that is your final answer.

Robert Daugherty · Answer

section 3.6 number 49. So I&#x27;m dealing with derivatives that require me to use the chain role. So here, I&#x27;m asked, defying the derivative. So why prime would be equal to the derivative of a co sign function that&#x27;s gonna be minus the sine function so e to the minus data squared. Now I have to take the derivative of what&#x27;s inside that parentheses, either the minus they to square. The derivative of the exponential function is just going to be the same function now, times the derivative of this expression right here. So the derivative of that is gonna be minus two data. So this final answer why Prime is equal to get to minus signs. They cancel. So it&#x27;s two data times e to the minus. They&#x27;d have squared times the sign of e to the minus, data squared, and that&#x27;s your final. It&#x27;s or tooth data E to the minus. Um, they have squared attempts, a sign of e to the minus state of squared

Robert Daugherty · Answer

section $3 6 Problem number 27 here were asked to find derivatives malfunction that involved, as needed to know the chain role. So here I need to find the derivative of our so our prime with respect to theta. So the first thing I&#x27;ll use will be the power rule. So this is negative. One times coasts, he can fade up, plus coach engine of failure to the negative to power times the derivative of everything that&#x27;s in this parentheses. So the derivative of the coast seeking is going to be, um this could be what, um, minus co tangent X Cosi connects. The derivative of the co tangent is just simply minus co seeking squared of X. Now, here&#x27;s the thing. Calculus is all complete. We&#x27;re into trig and algebra, usually at this point trying not to make mistakes with either of those. So what do I have here? So it looks like if I put everything together, um, So it looks like I&#x27;ve got um looks like I&#x27;ve got co tangent X co seeking X plus co seeking squared of X. All of that over Cosi can. Sorry, I&#x27;ve mixed up things. Just go back. Um everything is in terms of fatal. So nothing was wrong with the trick part, which is to be precise contends that data cause he can&#x27;t fade up minus co think squared of data. So everything is in terms of the terrible thing. Eso simplifying this. This becomes less. You have got a negative one that could be factored out here with the negative one. That&#x27;s gonna be positive. You&#x27;re gonna have co tangent. Fada Coast, seeking of fado plus CO ck in squared of theta over co seeking of Fada was continued of fada squared in this numerator I could factor out a coast seeking of Fada and then you&#x27;re left with what co changing of data plus coast Seeking of data over co seeking of data, it&#x27;s co tension of data squared. And now what? You look at this particular quantity co change a plus co ck. You&#x27;ve got two in the denominator one in the numerator That&#x27;s going to simplify to just finally co seeking of data over co seeking of Fada plus Co changing of data, which is your final answer

Runpeng Li · Answer

Okay, So for a problem, 21 were given our to be equal to To Over or finest Dita. So di archy feta will be now considered, um, conceded are to be two times or minus feta to the power of connective 1/2. So the directive will be two times. Look, dear one ab times four minus Tita to power up neck too. That&#x27;s negative. Three over too. And times remember, we have We also have to apply the changing role here. Now it is the director of the inside function. Well connected one. So, yes. So this gives or minus Sita to the power off negative rehab. Okay, Now we can see their data to be able to zero. So that means this will be for cute in the denominator one in the numerator. And we also need to take the square root. So that&#x27;s one over 64. That gives 1/8

Robert Daugherty · Answer

section $3 6 Problem number 50. Everything in this section deals with finding derivatives of functions that involve knowledge of the chain role. It also requires knowledge of product roll quotient rule, other things that you know it helps to think of this problem. You know, our product Rose only deals with two functions. So let&#x27;s think of it like this. I&#x27;ve got the function you see in brackets status squared, eat of the money to Data Times Co sign of five things. So let&#x27;s think of this with think of that as f times G and we&#x27;re finding is derivative to be f prime G plus g prime f. So let&#x27;s find the derivative of that first expression. So to find the derivative of the first expression. Well, that in itself is the product, Ruelas. Well, so look at that first term, which is state accused to think of. This is two things that are multiplied. So the derivative of Fada cubed is three. They squared times e to the minus two data plus, and then you&#x27;ve got, um, the derivative. So then you&#x27;re gonna have the derivative e to the minus two theta that&#x27;s gonna be minus two you to the minus two theater times Theda cubed. So that&#x27;s the derivative of the first expression, said the derivative of the first expression Times Co sign of five data. Plus that first expression, Which is why the fate a cube e to the minus two theta and the derivative of the co sign of five day That&#x27;s going to be minus five. Sign of five favor. So now lets you see what we can simplify. So it looks like I&#x27;ve got in all of these. There is an E to the minus tooth Ada Ah, and every term And it looks like there&#x27;s at least, um, uh um is there a fate a squared in every term? Yeah. There&#x27;s a data cube, Theda Cubed Data Cube. So it looks like if I factor out what&#x27;s in common, why prime is going to be data squared E to the minus tooth ada. And then what am I left with? I&#x27;m left with, um, this is going to be three. So you&#x27;re gonna have three, um, minus two theta. So you have three minus two theta co sign of five favor, and then when you factor out the fate of squared E to the minus two theta. You&#x27;re gonna be left with minus five data. So minus five data sign of five favor. And that should be about as close as we can get. So why promise? State of Squared either the miners tooth data times three months too late. Because I&#x27;m Fifita on this five day, the sign of five data. So, again, the key here was nothing really exotic, was it Was the product rule used twice and the chain role. So combining those rules together to get to my final answer for this derivative

Problem

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

Problem 28 Easy Difficulty

Find the derivatives of the given functions.
$$r=0.4 e^{2 \theta} \ln \cos \theta$$

Answer

$\frac{d r}{d \theta}=-0.4 e^{2 \theta} \tan \theta+0.8 e^{2 \theta} \ln \cos \theta$

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Catherine R.

Anna Marie V.

Kristen K.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

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$\frac{d r}{d \theta}=-0.4 e^{2 \theta} \tan \theta+0.8 e^{2 \theta} \ln \cos \theta$

Basic Technical Mathematics with Calculus

Catherine R.

Anna Marie V.

Kristen K.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

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

Catherine R.

Anna Marie V.

Kristen K.

Joseph L.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus