solve-the-given-problems-by-using-implicit-differentiation-at-what-points-is-the-tangent-to-the-curv

Question

Solve the given problems by using implicit differentiation.
At what point(s) is the tangent to the curve $y^{2}=2 x^{3}$ perpendicular to the line $4 x-3 y+1=0 ?$

Heather Zimmers · Accepted Answer

we&#x27;re going to find the equation of the tangent line to the curve y equals square root one plus X cubed at the 10.23 The slope of the tangent line will be the derivative at that point. So let&#x27;s start by finding the derivative. And I&#x27;m going to rewrite why, as one plus X cubed to the 1/2 power. And so then I can use the chain rule, starting with the derivative of the 1/2 power function. Bring down the 1/2 and raise one plus X cubed to the negative 1/2 power. And then we multiply that by the derivative of the inside, the derivative of one plus x cubed is three x squared. Now we&#x27;re evaluating this derivative at X equals two. We could simplify first, or we could substitute the two in first and simplify after. So that&#x27;s what I&#x27;m going to do. So 1/2 times one plus two cubed to the negative 1/2 times three times two squared. Okay, so this gives us 1/2 times nine to the negative 1/2 times 12 and nine to the negative. 1/2 is 1/3 so we have 1/2 times 1/3 times 12 So that is just too. So are slope is, too. Now we can use point slope form. Why minus y one equals m times X minus X one. To find the equation of the line where the point x one y one is 23 So we have why minus three equals air slope to times the quantity X minus two and we can simplify that and we&#x27;ll have the equation of the line. So let&#x27;s distribute the two. So why minus three equals two X minus four and then at three to both sides and we have y equals two x minus one.

Nicholas Barvinok · Answer

we have the equation x to the power of 2/3 plus why to the power of 2/3 equals four and we want to find the equation of the tangent line to this graph at the point minus three. I swear it of three one. So if we use this one of the this particular standard form of an equation for a line, why minus y one equals m X minus X one where M is the slope of the line and explain why one has a point on the line, then all we need to be able to write down the equation of this tangent line is a point that is going through which we already have and the slope. So all we need to find is the slope, meaning you want to find d Y over DX at the point minus three, Route 31 it was No, There should be no common there. Minus three times were three one. So in the way we&#x27;re going to find d white over DX. At that point is my first finding the wire RDX in general, which will do by implicit differentiation, which we will start by taking the derivative of both sides. Both sides of this equation with respect to X keeping in mind that why is a function of X so the derivative of X to the 2/3 we get 2/3 X by the power rule to the minus 1/3 plus Now we want the derivative of white, the 2/3 of the for this We&#x27;re going to use the chain rule. We have this inter function why and an outer function of ah inter function being raised the power of 2/3. So by the chain rule we start with by taking the derivative of the outer function which one for which we use the power rule and keeping the input the same and then taking the derivative of the inter function and multiplying by that do you? Why over DX? And now finally, on the right hand side, the derivative of four with respect to x zero, because it is Constance. So now what we want to do is solve for D Y over DX. So the first thing I&#x27;m gonna do is I&#x27;m going to move this first term over to the right hand side so that the left hand side is only my term. It has a d y over the accident and then we&#x27;re going to divide out by 2/3 wide to the minus 1/3. So we get d Y over DX equals minus 2/3 X to the minus 1/3. Then we&#x27;re so we can do this in steps. So we multiply by three halves which actually turns this into a minus one and then we can multiply. Why? To the minus 1/3 is the same thing is one over white with 1/3 so we can multiply by. Why? To the 1/3. And so what this actually gets us is minus. Why? To the 1/3 divided by X to the 1/3. So now let&#x27;s plug in our point. Do you? Why? Over DX and minus three Ruth 31 is well, why is one so the cube root of one is one. So I get minus one on top and the cube root of minus three squared of three is minus squared of three. Because if I cube that I get minus squared of three three, minus three squared of three. So this is one over. And so now we just plug in the relevant information to our formula. Why minus one equals one over square root of three X plus three squared of three and we&#x27;re done.

Ernest Castorena · Answer

creation of the Tangent line at the fourth is he Quincy Times X squared. My SMI squared at the 0.1 Commons query 3/2 and see if that&#x27;s current. Yeah, and now where you&#x27;ll find way prime illicitly So we&#x27;ll have four x cubed on the left side of the equation and then we&#x27;ll have, Let&#x27;s see, eight X that&#x27;s untracked on sea a white times. Why prime apart into the change room on, then? Yeah, where X cubed of minus a over a negative a blank is equal to y Prem, and we can get rid of this divide everything. But for so too to just the one here. And so we have that white prime at one square, 3/2 is equal, Teoh One mine intends to all over. Then get it squared three because they&#x27;re all the time, not by two. And so we have one over this query to three. And well, the equation for Attendant Lynn will be why I minus the square root of 3/2, is equal. Teoh, our slope square the 3/3 times X minus one. And so we have. Why is equal to the square root of 3/3 times X, and then we have minus square of three over. Let&#x27;s get the common denominator, so we&#x27;ll have six over to plus three Square E 3/6. And so we have the equation. Square area 3/3 time. Why is equal to Scary 3/3 times X plus? That&#x27;s very of 3/6, all right.

Amy Jiang · Answer

Okay, so we want to find the values of which the tango line is given to be perpendicular to wise vehicle checks. So let&#x27;s recall that if we&#x27;re perpendicular to a falling graph core line our slope in this case I slept is one well are perpendicular. Soap is going to be equal to one over negative one or negative one over this time here. And this is our perpendicular. So which is negative one? So here we have that. Do you? Why, Over the X is equal to negative one. Okay, so let&#x27;s find a derivative of our white term and set that equal to one. So we&#x27;re gonna get we&#x27;re gonna use a question rules. That&#x27;s one times experts to minus exports. Three such minus X minus three all over. Exposed to to divert to. So this is equal to a negative one over X push to depart to when is this equal to negative? What? Okay, so it&#x27;s multiply both sides by X plus two Did a part two. So what&#x27;s exports to time, Jack Sports too. So we get X squared was to expose to accidents for exports, for you have negative one is equal to negative. X squared minus four X minus four Woods, move all over terms Your left hand side Do you get X squared plus or X plus four minus one? So that&#x27;s supposed today is equal to zero. So this time we have sex. Um, in fact, that is exposed. Three. Well, let&#x27;s use our quadratic formula. Do we have X is equal to negative B plus minus the square root of B squared minus four times aimed fine. Three. All over two A. So, what? 16 minus four times three. That&#x27;s a four. So we get the square root of four, which Jews who have minus four plus minus to all over to to the contemplative itis into a minus to post minus one. So we got a negative one or a monastery, so these are our two values for X.

Ernest Castorena · Answer

give us the face. Cute y squared is equal to negative three, and they want us to find the tangent line equation at the point. Negative one. Come on and get it. Three and so Product draw will get three x squared times y squared, plus execute times two y in times why a print is equal to on other product. Role will have negative for your life, plus negative three x y prime. Okay, so we can collect our wife prime terms on the same side of the equation. We&#x27;ll have to excuse point. What&#x27;s The Yanks is equal to bullets minus three y minus X squared y squared. And then we can divide both sides of the equations of this factor. Teoh Ex Cute by class three eggs so that are derivative about the point. And they could have one common negative thing is equal to negative green. Negative minus Think negative. One squared. Three squared all over. Let&#x27;s see two times negative. One. Cuban is making one. Sometimes negative 33 times, negative one. Ultimately, we have nine. But see you subtract 27 because that&#x27;s nine times negative all over. I see let&#x27;s girl down on it all over six. Subtracting no Mrs Thyself in that all over a six track three, which is just three. So then we have negative 18/3, which was just negative. Six. So the equation are tangent. Line is why plus three is equal. Teoh Negative. Six times X plus one. Well, on the right hand side, we have negative six and track six. And then when you subtract three on both sides of the equation, to get white is equal to negative six x minus tonight. All right?

Heather Zimmers · Answer

the equation of the tension line to the curve. You know, we find a derivative by using Chin Russo. This is outside derivative on the inside the room to ISS React square by the power room. So but actually hope for your evaluation that you ever supposed to That gives a slope because three times for she square of the one plus eight, So square with nice three, three, two, six and have twelve over six is two. So we have a slow beat was too. And the point here. So he was a point slow formula to figure out the equation of the tension line. So this should be why minus three, most two times X minus two. And ofcourse, you can simplify it if you want.

Problem

Solve the given problems by using implicit differ…

Problem 32 Easy Difficulty

Solve the given problems by using implicit differentiation.
At what point(s) is the tangent to the curve $y^{2}=2 x^{3}$ perpendicular to the line $4 x-3 y+1=0 ?$

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