Log in 20leunge 7 years agoPosted 7 years ago. Direct link to 20leunge's post “I'm so confused, where di...” I'm so confused, where did t come from? How do we know x and y are functions of t? • (12 votes) kaikycintra 3 years agoPosted 3 years ago. Direct link to kaikycintra's post “We know because there are...” We know because there are derivatives of x and y with respect to t, if they weren't functions of t, then such derivatives would have no value. (6 votes) Matthew Chen 5 years agoPosted 5 years ago. Direct link to Matthew Chen's post “Could you also solve this...” Could you also solve this equation using implicit differentiation? • (8 votes) Emma a year agoPosted a year ago. Direct link to Emma's post “Using implicit differenti...” Using implicit differentiation: Take the derivative of both sides (note that we are taking dy/dt, not dy/dx, because we are taking the derivative in terms of t as the question calls for): When dy/dx is multiplied with dx/dt, we get dy/dt. Since we are finding dy/dx when x is 9, we get: Basically, this is what is shown in the video, but in a more direct format. (5 votes) J.R. Foster 4 years agoPosted 4 years ago. Direct link to J.R. Foster's post “Where did the dx/dt come ...” Where did the dx/dt come from? How does that relate to the chain rule? If we did the chain rule on sqrt(x), that would come out as 1x^-1/2 and that would be it, wouldn't it? Why tack on this extra dx/dt and why is it there? Does it have something to do with time being a factor in this equation? • (3 votes) loumast17 4 years agoPosted 4 years ago. Direct link to loumast17's post “Sal mentions that the pro...” Sal mentions that the problem states that x AND y are differentiable funtions, so x is also a differentiable function, which means x is a function. the problem then says dx/dt is 12 so that is basically giving us the answer that x's independent variable is t. so you can think of y as y(x) or y of x and x as x(t) or x of t. it relates to the chain rule because we are saying y is a function of and x is a function of t so it is one function inside of another, which is what the chain rule deals with. We don't use the chain rule here as we normally do. instead we fill in the blanks. the chain rule says dy/dt = dy/dx * dx/dt. we don't know what x(t) is though so we cannot solve for dx/dt, but the problem gives us what it is so we can just fill it in. time being a factor is often why you would have a problem like this, but not always, the big takeaway is just how to take derivatives of composite functions that are not written out in the standard way. Let me know if something didn't make sense, or yous till have questons. (3 votes) jeremy.yuan1 4 years agoPosted 4 years ago. Direct link to jeremy.yuan1's post “I don't understand how dy...” I don't understand how dy/dx becomes (1/2)x^(-1/2) instead of 1/sqrtx • (3 votes) loumast17 4 years agoPosted 4 years ago. Direct link to loumast17's post “sqrt(x) can be written as...” sqrt(x) can be written as x^(1/2) (1/2)*x^(-1/2) negative exponents can be written as x^(-a) = 1/(x^a) so we can rewrite the answer we got as this (1/2)*1/(x^(1/2)) then as we said before x^1/2 can be written as sqrt(x) (1/2)*1/sqrt(x) then we just multiply the fractions 1/(2sqrt(x)) so 1/sqrt(x) is actually pretty close, just need to keep in mind it is also being multiplied by 1/2 Let me know if that did not help. (4 votes) Garret Cervantez 7 years agoPosted 7 years ago. Direct link to Garret Cervantez's post “Can someone tell me if I ...” Can someone tell me if I understand this correctly in regards to dx/dt. After re-watching the video I am less clear than a I was a few minutes ago. dx/dt is saying what is the derivative for the FUNCTION x when you insert the PARAMETER t. Thanks for the help. • (2 votes) Zach 7 years agoPosted 7 years ago. Direct link to Zach's post “dx/dt is an expression me...” dx/dt is an expression meaning "the derivative of x with respect to t". So it's as you said: it's the derivative of the function x when using the variable t. (5 votes) Shivam 7 years agoPosted 7 years ago. Direct link to Shivam 's post “Shouldn't we have 2answer...” Shouldn't we have 2answers for this question i.e. 2, -2 we want to calculate dy/dt for x= 9 and we know x-y relation so we get y = +3,-3 for which we have to calculate dy/dt since y = x^.5 , so x= y^2 given is, dx/dt = 12 substitute two values of y( which we found at top) in this equation. we get • (2 votes) tyersome 7 years agoPosted 7 years ago. Direct link to tyersome's post “If you are given a square...” If you are given a square root, you should presume it to be the principal (i.e. positive) square root. If you introduce a square root, you need to consider both the positive and negative roots. (4 votes) Leon a year agoPosted a year ago. Direct link to Leon's post “i used implicit different...” i used implicit differentiation and solved the problem. i don't get what's new and important being taught that we need to be concerned about learning in this video. • (2 votes) Venkata a year agoPosted a year ago. Direct link to Venkata's post “Nothing new then. Assumin...” Nothing new then. Assuming you've differentiated w.r.t t, that's what they want you to do here. They want you to get accustomed to having independent variables that aren't x and y. (4 votes) veratility 8 months agoPosted 8 months ago. Direct link to veratility's post “Is this just the chain ru...” Is this just the chain rule? In the video dy/dt is defined as dy/dx * dx/dt. The chain rule for dy/dx is defined as dy/du * du/dx, where u is the inner function of y. Not sure if there is a gap in my understanding. • (1 vote) dougw 6 months agoPosted 6 months ago. Direct link to dougw's post “Yes, this is an applicati...” Yes, this is an application of the chain rule via implicit differentiation. You may want to review implicit differentiation. Khan Academy has a lesson on it. The other area where you may have a misunderstanding is in the application of the symbols. The chain rule does not require the outside function be set to y, the inside function be set to u, and that the independent variable (i.e. the argument) of the inside function, u, be set to x. The chain rule only provides a method for finding the derivative of a composite function. The functions of the composite can use any symbols that they wish or that you wish to assign to them. (3 votes) jonnuki 2 years agoPosted 2 years ago. Direct link to jonnuki's post “Would have been way clear...” Would have been way clearer to simplify to 1/2sqrtx (dx/dt) Which is 12/2sqrt{x}. Evaluating at x = 9 • (2 votes) Yaotian Zhang 2 years agoPosted 2 years ago. Direct link to Yaotian Zhang's post “I got different answer so...” I got different answer somehow. • (1 vote) Jerry Nilsson 2 years agoPosted 2 years ago. Direct link to Jerry Nilsson's post “𝑑𝑥∕𝑑𝑡 = 12 ⇒ 𝑥 = 12�...” 𝑑𝑥∕𝑑𝑡 = 12 ⇒ 𝑥 = 12𝑡 + 𝐶 𝑥 = 9 ⇒ 12𝑡 + 𝐶 = 9 ⇒ 𝑡 = (9 − 𝐶)∕12 𝑦 = √𝑥 = √(12𝑡 + 𝐶) Plugging 𝑡 = (9 − 𝐶)∕12, we get (2 votes)Want to join the conversation?
y=sqrt(x)
dy/dt = (1/2 x^(-1/2))(12)
where (1/2 x^(-1/2)) is dy/dx and 12 is, as given, dx/dt.
dy/dt = (1/2 (sqrt9)^(-1/2))(12)
dy/dt = (1/2 * 1/3)(12)
dy/dt = (1/6)(12)
dy/dt = 2
d/dx x^n = n*x^(n-1). Here n =1/2 so let's plug in.
Explanation:
we substitute x with y^2 so above equation becomes d(y^2)/dt = 12
so, applying chain rule and simplifying we get,
dy/dt = 6/y
dy/dt = +2,-2
You might want to review the material on radicals:
https://www.khanacademy.org/math/algebra/rational-exponents-and-radicals/alg1-radicals/v/introduction-to-square-roots
12/2sqrt{9} = 12/2*3 = 12/6 = 2
dx/dt = 12, get x(t)=12t,
then when x = 9, t is 3/4.
y = root x, so y is root of 12t,
so dy/dt is 1/2*(12t)^(-1/2), plug in t=3/4, the answer is 1/6.
Please tell me where I did wrong.
(𝐶 will cancel out with itself later on, but for now we should have it there)
⇒ 𝑑𝑦∕𝑑𝑡 = 1∕(2√(12𝑡 + 𝐶))⋅12 (chain rule!)
= 6∕√(12𝑡 + 𝐶)
𝑑𝑦∕𝑑𝑡 = 6∕√(12(9 − 𝐶)∕12 + 𝐶) = 6∕√(9 − 𝐶 + 𝐶) = 6∕√9 = 6∕3 = 2
Video transcript
- [Instructor] We are toldthe differentiable functions X and Y are related bythe following equation, Y is equal to square root of X. And it's interesting, they're telling us that they are bothdifferentiable functions, even X is a function, must bea function of something else. Well, they tell us thatthe derivative of X with respect to T is 12 and they want us tofind the derivative of Y with respect to T when X is equal to nine. So, let's just make surewe can understand this. So, they're telling us thatboth X and Y are functions. Arguably they are both functions of T. Y is a function of X but then X is a function of T so Y could also be a function of T. One way to think about it is if X is equal to F of T, then Y is equal to the square root of X which would just be F of T. Another way to think about it if you took T as your input into your function F, you're going to produce X and then if you took that as your input into the square root function you are going to produce Y. So, you could just view thisas just one big box here, that Y is a function of Y but now let's actuallyanswer their question. To tackle it we just haveto apply the chain rule. The chain rule tells usthat the derivative of Y with respect to T is going to be equalto the derivative of Y with respect to X timesthe derivative of X with respect to T. So, let's apply it tothis particular situation. We're gonna have the derivative of Y with respect to T is equal to the derivative of Y with respect to X. Well, what's that? Well, Y is equal to theprinciple root of X. You could also write this as Y is equal to X to the one half power. We could just use the power rule. The derivative of Y with respect to X is one half X to the negative one half. So, let me write that down. One half X to the negative one half and then times the derivative of X with respect to T timesthe derivative of X with respect to T. Well, let's see, we wanna findwhat we have here in orange, that's what the questions asks us. They tell us when X is equal to nine and the derivative of X with respect to T is equal to 12. So, we have all of the information necessary to solve for this. So, this is going to be equal to one half times nine to the negative one half, nine to the negative one half, times DX/DT, the derivativeof X with respect to T is equal to 12, times 12. So, let's see, nine to the one half would be three, nine to the negativeone half would be 1/3, so this is 1/3, so this will all simplifyto 1/2 times 1/3 is 1/6, so we could have a six in the denominator and then we are going tohave a 12 in the numerator. So, 12/6, so the derivativeof Y with respect to T when X is equal to nine and derivative of X withrespect to 12 is two.
