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Find $ dy/dx $ by implicit differentiation.$ \sin (xy) = \cos (x+y) $

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$$y^{\prime}=-\frac{y \cos (x y)+\sin (x+y)}{x \cos (x y)+\sin (x+y)}$$

01:21

Frank Lin

02:29

Doruk Isik

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 5

Implicit Differentiation

Derivatives

Differentiation

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

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.

03:29

Find $ dy/dx $ by implicit…

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01:15

01:38

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02:53

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00:09

Find dy/dx for the followi…

So for this problem, what we're gonna want to do is implicit differentiation and implicit differentiation is very good when we have some expression with, you know, an X and A Y, and it's equal to another expression. So we have this equation. Um, that's not just a simple asses. Why equals something? It's some function of X and Y perhaps and then another function of X and Y on the other side. So that's why we want to use implicit differentiation. So what we're given here is thesis ein of X y being equal to the co sign of X plus y. We're gonna differentiate both sides with respect tax. So it's gonna look something like this. So now when we do this, what we end up getting is we want to do the chain role. So, with the chain role we now have that this is the co sign of X y times. Why plus X y prime and that's doing chain role. That's gonna be equal to a negative sign of X plus y times one plus why prime? So with all of this, we want to simplify further. So we have now is why cosine X y plus x co sign X y times y prime And that's gonna be equal to the negative sign of X plus y minus the sign of X y x plus y white prime. So now that we have all this, we want toe ad, um, this right here to both sides and then we want to factor out the why prime. So when we do that, we get, um, X curse Sane X y plus sign X plus y We moved this over here and we moved this over here. So now that we have that, this is going to be all times. Why prime? And it's equal to a negative sign. X plus y minus. Why co sign X y and then we just divide the whole thing by this portion right here. So what we end up getting is that why prime is equal to a negative sign? Plus why, minus why could sign ex wife all over x co sign X y plus sign X plus y. This will be our final answer for the problem

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