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Problem 37 Medium Difficulty

Find $ y" $ by implicit differentiation.
$ \sin y + \cos x = 1 $

Answer

$$
y^{\prime \prime}=\frac{\cos x \cos ^{2} y+\sin ^{2} x \sin y )}{\cos ^{3} y}
$$

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Video Transcript

both probably want to find the second derivative using implicit differentiation. So we're given in this case is sign. Why puts coastline ax equals one and the main reason we use inputs a differentiation is if we ever have functions in which the X and the Y are together. And there's not just one derivative with sacred to take the derivative of both sides because we're given an equation. Uh, if we're taking the derivative of an expression, that's different. But since we have this equation here will take the derivative both sides. When you take the derivative of the right side, it's just gonna be zero, since one is constant. The left side, however, what will end up having is the sign y the derivative that would be co sign y. But it'll be time to wipe crimes and sort of taking the derivative with respect to X. And then it will just be minus sine X. Since we're taking the derivative with respect to X, so then what we'll have is waas. Add the sign next to this side, divide coastline y over, and what will end up getting as a result is that why prime is equal Thio sign of acts over coastline of wine. Then we want to find why Double prime. So we'll take the derivative of that right there. When we do that, we see that we will use the quotient rule. So why double crime is going to be equal to the low times the derivative of the high. So the denominator times the derivative so coastline Why times the derivative of the numerator, The derivative of the numerator is, uh, going to be the coastline of acts. We get coastline next time coats on why minus the sign of acts. I'm the derivative of the denominator. The derivative of the nominator is going to end up giving us, uh, this will be a negative sign y times. Why prime? And then all of this is going to be over the denominator squared. So it'll be over cosine squared. Why, um then when we substitute all this in what we see is that we have coastline Next times coastline y minus sine x. These negatives will cancel right here, so we'll just get plus sine x times sign why? And then we know that this right here is equal to the sign of acts over the coastline of y. So that would just make this cubed right here. And it'll make the sign of ax squared so we'll just do this. This will be our final answer for, um what we have. The only other thing that we would have to do is make sure that the cosine of why squared as well, because we need Thio. Multiply everything through. This would be our final answer for the second directive.