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Find $ dy/dx $ by implicit differentiation.$ \cos (xy) = 1 + \sin y $

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00:30

Frank Lin

01:34

Doruk Isik

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 5

Implicit Differentiation

Derivatives

Differentiation

Missouri State University

Campbell University

Oregon State University

University of Michigan - Ann Arbor

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.

03:51

Find $ dy/dx $ by implicit…

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02:55

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05:00

So for this problem, we're gonna be doing implicit differentiation, which is a very useful technique when dealing with multiple variables in a function or wise and excess all throughout a function. So we have in this case is that because sign of X Y is equal to one plus the sign of wine. So we want to perform inputs, differentiation. And to do that, what we're going to dio is take the derivative this side and then take the derivative of this side. First, we'll take the derivative of the left side. Um, when we take the derivative left side, we remember that the derivative of cosine is negative sign. So what we'll have here is negative sign X y But then we have to do the chain role. So that's going to give us Times X. Uh, why prime? It's gonna be the product role. So x y prime plus why the x t x, which is just that which is why then we'll have that this is equal to the one will go away. Since it's the driven of a constant andan, all we'll be left with is the co sign of why and why prime So this is what Well, have right here and then we'll do next is we're going Thio Um we're going thio Add x sine x y to both sides and the reason why we have the negative x sine x y is because this right here is distributed. So, um, what will have in this case since this is all distributed is X It's a negative X sign fine times like crime. And then that's minus sign of X Y yeah, on Ben, we also want tohave. This is minus Why sign of ex wife? Just thio confirm that then what will have as a result is since this was all way, have to keep mine. This is all to be I distributed with So this is gonna be equal to the same stuff over here then our goal is to ultimately isolate the UAE prime. So we're going toe Add x sine x y y prime on both sides. So when we do that, we end up getting this right here. So it's positive announced in a negative. We can factor out the why prime on, we'll get a positive here. So be plus all of this and then lastly we can just divide this whole thing right here. When we do that, we end up getting this as our anthem for white crime. It's negative. Sign X y over coastline y plus X sine x Y, um, and that's our D Y DX and our final answer.

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