23–51. Calculating derivatives Find the derivative of the foll...

Question

23–51. Calculating derivatives Find the derivative of the following functions.

y = sin x / 1 + cos x

Accepted Answer

Welcome back, everyone. In this problem, we want to find the derivative of Y E equals x divided by 2 minus X. A says it's 1 minus 2 x x divided by 2 minus X X squared. B 2 x divided by 2 minus X X squared. C negative s x divided by 2 minus X2, and the D negative X divided by 2 minus X2. Now notice that Y is a caution. What do we know about finding the derivatives of cautions? Well, recall that we can use the quotient rule of differentiation. It basically says that if we have a function y that's equal to u divided by V, where U and V are functions of X, then the derivative of Y will be equal to u V minusuv divided by V2, where U and V are the derivatives of U and V respectively. Now for our quotient, our numerator is X, so U equals X, and our denominator is 2 minus sine X. So that will be V. If we can differentiate our numerator and denominator with respect to X, then we should be able to apply the quotient rule. So let's go ahead and do that. So we know that for our numerator, the cosine of X, OK, then the derivative of the cosine equals negative X, OK? So U is negatives X. Next for our denominator V, which equals 2 minus sine X. To find a derivative, we can differentiate both of those terms separately, where the derivative of 2 is 0 and the derivative of sin X is negative, uh, the derivative of sin X is class X. So that's 0 minus X, which is just going to be class X. So, sorry, negative class X, my apologies. So now that we have the derivatives of UV, and sorry, we have UV and its derivatives, then we can apply the quotient rule. Because no, that means the derivative of Y then is going to be equal to UV. So that's negative sin X multiplied by 2 minus sine X. Minus UV, so that's the cosine of X multiplied by the negative cosine of X. And now all of this is being divided by V2 minus sin X2. So now we found the derivative. Let's try to simplify. Let's simplify in our numerator. So now we can distribute negative sin X for our first product, which is gonna be -2 X, OK, plus sine square x, OK? I know for our second term, class X multiplied by negative. Cos X is negative cost 2 X. And when we subtract that, it's gonna be plus squared X, OK? And then we can rap back our denominator. Now we can simplify this even further because in our numerator it has evidence of the fundamental trigonometric identity. Recall that that identity tells us that the square of the sign plus the square of the cosine for the same argument is equal to one, OK? And in this case, notice here we have sin squared X plus cosine squared X. So if we apply the identity here, then we could rewrite this as 1 minus to X, OK? All divided by 2 minus X or squared. Thus, this is the derivative of Y. If we look back at our answer choices, that tells us then that A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

23–51. Calculating derivatives Find the derivative of the following functions.
y = sin x / 1 + cos x

Key Concepts

Derivatives

Quotient Rule

Trigonometric Functions

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