solve-the-given-problems-by-using-implicit-differentiation-two-resistors-with-resistances-r-and-r2-a

Question

Solve the given problems by using implicit differentiation.
Two resistors, with resistances $r$ and $r+2,$ are connected in parallel. Their combined resistance $R$ is related to $r$ by the equation $r^{2}=2 r R+2 R-2 r .$ Find $d R / d r$.

Raushan Kumar · Accepted Answer

we are going to do problem number 45 on this question. 32 decisions are joint. Okay. Are we in on our do? So they are joined in parallel, so the expression is given for the resistance. That is run by articles two and by r one plus one by our two. Okay, so it is given that are one is three. Um, on It is also given that that is equal interest distances to So we have to find the value off rt. Okay, we have to find the value of party. So that is just put here. We put area this 11 by two, we put the value Polonia. This will even by three plus one by our two. Okay, so we compare this says one by our tow, this will be equals toe one by two manners one by three. This will give us the value off when we are too. So here we would have won by our two. This will be our six that this is three managed to. So we will get her one by are two physicals toe one by six. Okay, so from here we can write that are too is equals 264 So this is the answer. That&#x27;s so Thank you

Karl Schaefer · Answer

and this problem we&#x27;re asked to find e y d x and the second road of de spread wide DX squared. Given the equation X Y plus widespread equals one. So first, to find the first derivative, we use implicit differentiation. Just take the derivatives of both sides. So the Druid of X times why we need to use the product rule. So the drift of the first times the second plus the first times the derivative of the second. Now we want to add to this the derivative of why squared So here we have to use implicit differentiation. Kind of think of it like a chain rule. So the drift of why I squared is too. Why do you want the ex? We&#x27;ve kind of done this and in previous examples, Finally, on the right side the drift two of one is zero. Now we want to get all the d Y. T X is on one side, so let&#x27;s keep them on the left side Extra wide E X plus two. Why do I d X equals minus y? So all we&#x27;ve done here is we&#x27;ve moved is why it&#x27;s attracted to the other side and the reason why we do this is so that we can factor out a d y DX on this side. And then when we divide minus why then when we divide by X Plus two Why we get our expression for D y d x d y d x is equal to minus y over X plus two. Why now? We want to find the second derivative from open up a new page for that. So we start with our first derivative again. Do you Y d x equals minus y over X plus two I now, to find the second review, we just take the derivative of the first derivative. So the second derivative Well, we see that we have to do a chain rule here. Shoot me. Not a general question rule So low d hi, Minus Hi, Di Lo. So the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator all over the denominator squared. At this point, we&#x27;re almost done. But we&#x27;re asked to have our answer on ly with exes and wise. But here we have G Y d X, which is not in terms of X and y But luckily we have a formula for D y d X in terms of x and y that we can use. Just plug in. So let&#x27;s go ahead and do that. X Plus two. Why times minus minus. Why over X Plus two. Why now I&#x27;m going to combine these two minuses to a plus here, minus and minus is a plus. Plus why? Times one plus two times minus y over X plus two y all over X plus two y squared. So this is is a perfectly good answer, but we should kind of let the dust settle a little bit, you know, cancel some some stuff for there&#x27;s obvious cancellation. So, for example, here we have two minuses. Let&#x27;s make those a plus. And now we have an X Plus two wine. The numerator, which cancer with an X plus two wine. The denominator. So we have Why, plus why? So let&#x27;s go ahead and distribute the Why here. So why, plus why? Plus two times negative y squared over X plus two. Why all over acts plus two. Why squared now while we&#x27;re here? Let&#x27;s go ahead and combine the numerator into a single fraction so why Plus why is, too why now we want to make a common denominator so we could get a single fraction in the numerator while we&#x27;re here. Let&#x27;s go ahead and just put a minus to appear and then get this two up into the numerator. So our common denominators X plus two. Why? So we have two wide times X plus two. Why minus two y squared all over X plus two wise, that&#x27;s our new numerator. But then we still have the X Plus two y squared in the denominator before. So here we just combine our numerator into a single fraction. Now let&#x27;s do one more step. Let&#x27;s distribute two acts. Why plus four y squared minus two y squared all over. Let&#x27;s combine our two denominator to a single denominator so we&#x27;re dividing by X plus two Wine X plus two y squared. So when we multiply those together, we get X Plus two White Cube. Our final answer for the second derivative is two x y plus two y squared over F plus two. Why cute

Tom Greenwood · Answer

We&#x27;re trying to find the second derivative of this equation. So let&#x27;s start with the product rule here. First function X Times the driven of other second derivative of Why is gonna be y prime or Dubai DX Plus the second function. Why times The derivative of the first drive of X is just one and then plus the derivative of two y squared, which will give us to i times. Why Prime, because of the chain room that equals the derivative of a constant, is zero skinning. So now it&#x27;s clicked our prime terms on the left side and let&#x27;s move our wide to the other side. Now let&#x27;s factor out the prime that gives us X Plus two y times y prime equals negative. Why, that means that why prime our first driven equals negative. Why over X plus two y. So there&#x27;s our first derivative. Now we have to find the second derivative, which means we have to now use the quotient rule with this expression. So why double prime equals bottom function? Next plus two y times the derivative of the top, which will be negative. Why prime minus the top function negative. Why times the derivative bottom, which will be one plus two. Why? Prime metal gets divided by the denominator X plus two y square. Right now we need to do whole lot of distributing, so distribute the negative white crime through here. Actually, let me change that. How about if we substitute first? Let&#x27;s just say distribute the negative. Why here? Let&#x27;s hold off on this. Why, Prime, they were in a substitute there. All right, so we have why Double prime equals X plus two y times. Negative, like prime. And now the minus and the negative here that I just become a plus. So distributing why through that have become plus why, plus two. Why? Why Price? And that&#x27;s all divided by the X Plus two. I squared. Okay, Now I&#x27;m going to just do my substitution. I&#x27;m going to replace my y primes here with are y prime that we had back here. So this will be X plus two y times. The negative of the negative will give me a positive. Why? Over X plus two y plus why? Plus two. Why times the negative. Why Over X plus two y And that&#x27;s all still divided by the X plus two. Why being squared? So the X plus two wives here will reduce. That just gives me why, plus another why. So I&#x27;ll give us two eyes minus the two y times the negative. Why will give us the minus two y squared over the X Plus two y, and that&#x27;s all still divided by the X plus two y. And now if we multiply top and bottom by X plus two y to get rid of the complex right complex reaction here we have to. Why? Times a quantity X plus two y minus two y squared over X plus two y should been squared. But now X plus two y is cute and so one last thing we can do. We can distribute the two why through here and that will give us a couple of like terms to combine. So we end up with our second derivative of being, and I ran out of some space. So bear with me and going to do it right here in the middle. Our final answer for the second drift, it will be two x y two y times to while give us four y squared minus two y squared will be plus two y squared, divided by the X plus two. I cute and there is our second derivative answer.

Tanishq Gupta · Answer

This is the solution for personal seven. No, Let us first sold for India. Question when it was one of our So I wake up on this, I went on this Be no easy to find the parts. Turn that off with respect to one. So then I I way off. That will be, uh you sweat. I went less I No, my art. I will get on Leicester Square. This flying assured that it will, uh, they respect. I want frigates nature by then I for LR by I want to be really Ben, we differentiate with respect, I No, we work our way. We see there. Why are one is current events evil and this implies that is raising off one It&#x27;s fixed. This implies that on the increase off our money our respects village When a reason we see that 5 10 are you? Is that this implies? That is I&#x27;m fixed. This is a place that decrees off when it&#x27;s really please. Oh, all right, So this final solution or question Thank you

Robert Daugherty · Answer

$3 7 Problem number 19 Implicit differentiation. So I&#x27;m given the equation of a circle X squared plus y squared equal one. And I&#x27;m asked to find a couple of things I&#x27;m asked to find e y d X. And then I&#x27;m asked to find the derivative of that So d y squared d X. So let&#x27;s start with the 1st 1 to find d y. The X means to differentiate this entire equation with respect to X. So this leaves me with two X Plus two. Why do you Why the X is equal to one. A difference of the right hand side. I get zero What I&#x27;m after here solving for D Y the X. So this tells me that D Y D X is equal to negative two. X divided by negative too. Why? So this tells me that D Y d X is equal to negative X over what? So that&#x27;s the first half finding D Y B X. Now the next part will be finally then the derivative, the second derivative of this. So starting with this D Y T X and taking its derivative with respect to X. So the second derivative of Why, With respect to X is going to be a I gotta differentiate negative X over. Why? With respect to X, that&#x27;s gonna be the quotient rule. So remember, with the quotient rule, it&#x27;s the derivative of the numerator times of their nominator minus that river, the nominators of June writer over the denominator square. So look at the, um, the numerator. Okay, so the derivative of the numerator is just going to be, um it&#x27;s gonna get me a negative negative one, OK, And then all of that terms, why minus? And then I&#x27;m going to have negative X times the derivative. Why, that&#x27;s just d Y d x All of this over. Why squared? So this becomes negative. Why? Plus, um x do I d x all of this over. Why squared now? You know, from our first, um, part of the problem is that D Y d X is negative. X over. Why? So this becomes negative. Why, plus X times negative X over Why all of that over? Why squared? So then we can get a common denominator and we can see that this is, um, minus y um, minus why squared and then minus expert over Why over y squared. So this is minus y squared minus X squared over. Why cubed So going onto the satyr Paige. So I&#x27;ve got d squared of Why the ex It&#x27;s equal to So let&#x27;s look back. Um, this is what minus y square minus X squared. So minus y squared minus X squared over. Why cubed? Another way of writing This is negative. X squared plus y squared over. Why cubed now? Let&#x27;s go back. What was our original X squared plus y squared was equal to what? One of the equation of this circle. So our final ants here is minus one over white. Cute. Okay, so that is the answer for a D To why DX the second derivative? Why, with respect to X. So again we found D Y d X with minus AA minus x over Why and the second derivative. So taking the derivative that with respect to X minus one over White Cube

Problem

Solve the given problems by using implicit differ…

Problem 41 Easy Difficulty

Solve the given problems by using implicit differentiation.
Two resistors, with resistances $r$ and $r+2,$ are connected in parallel. Their combined resistance $R$ is related to $r$ by the equation $r^{2}=2 r R+2 R-2 r .$ Find $d R / d r$.

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