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Find $ dy/dx $ by implicit differentiation.

$ x^2 - 4xy + y^2 = 4 $

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$$y^{\prime}=\frac{2 y-x}{y-2 x}$$

00:33

Frank Lin

01:10

Doruk Isik

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 5

Implicit Differentiation

Derivatives

Differentiation

Missouri State University

Campbell University

Harvey Mudd College

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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this problem, We're gonna be doing something What's known as implicit differentiation, which is very useful if we have multiple variables like y and X and a function. And we need to take the derivative of why with respect to X So we're given X squared minus four x y plus y squared equals four And what we do is we take the derivative of both sides. Now it's obvious that if you take the derivative of this site since the constant, it'll be zero eso. Now we want to focus on the derivative of this side right here. DDX So we know that it will be the same thing is taking the derivative this minus the derivative this plus the derivative of this. So what will end up getting as a result? Um, it's fairly clear here that what we'll get is, uh, two X, that's the derivative. Then we'll end up getting minus for why, But then, since we have, um, that right there, we'll have minus four. Why? And then Plus, uh, since we have the wire right here, what will end up having as a result is going to be a why prime, So multiply this by white crime. Then in addition, will have plus y squared. The driven will be two y I'm white prime So now what we have is two X minus four y minus for X Why prime? Because we want Thio ultimately have the change well done. So it will be minus four x times y prime plus two y times. Why prime? And like we said, that will be equal to zero. Then we can factor out the white prime here. So we do that. We end up getting a two x minus four y plus a negative for X plus two y with the UAE prime factored out equal to zero. Then we move over the two x and four. Why? So when we do that, we'll end up getting right here. But we have to switch the signs. So the plus for why minus two X Then we want to divide everything by this right here. So I'll take that great and divide it. We know that the queues can cancel out, so it will be a well cancel it out with a negative too. So this will be X minus two by over Xu acts minus wine. That will be our final answer for why Prime

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