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

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

00:30

Frank L.

01:34

Doruk I.

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 5

Implicit Differentiation

Derivatives

Differentiation

Missouri State University

University of Michigan - Ann Arbor

University of Nottingham

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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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