Find y'' for the following functions.
y = cos θ sin θ

Question

Find y'' for the following functions.

y = cos θ sin θ

Accepted Answer

Welcome back, everyone. In this problem, we want to find a second derivative Y for the function Y equals 14 Ts T. A says it's -28, sin 2 T B 28, 2 T C 14 2 T, and D 7 sine T. Now to find the second derivative of the function Y, essentially just means that we're going to differentiate Y twice. So let's find the first derivative, differentiate it again to get the second derivative. Now notice here that Y is a product. 14 cost T multiplied by a sine T. So to differentiate Y we can use the product rule of differentiation. What does the product rule say? We'll recall that the product rule basically tells us that if we have a function y that's equal to U and V, you multiplied by V sorry, where U and V are differentiable functions, then the derivative of Y. Equals U V plus UV. In other words, we differentiate both our terms and plug them into the product rule to solve for the derivative. So in this case then for our term, if we let you be because T and V besin T, OK. Then, let me put that in red, actually, we can differentiate both of those and multiply them by our constant. Leve our constant as the result, OK? So that means if you is Katy. Then U is a negative sine T. It's the derivative of the cosine. And if he is sin. Then V is Katy, OK. So now, in this case, we're gonna multiply them by each other. Which means that the derivative of Y is gonna be equal to 14, OK, multiplied by UV. So that's sine T. Multiplied by negative sinet. Plus UV. So that's cost T multiplied by T. Now here we could rewrite this as 14 multiplied by cost squared T minus sine squared T, OK. And now when we write it like this, we may know that it resembles the double angle formula, OK, or the double angle identity in trigonometry. Recall that identity basically tells us that the cosine of 2 X, is going to be equal to a cosine squared of X minus the sine squared of X. So in this case, it's telling us that the cosine of 2 T. Is going to be equal to cosine squared of T minus squared of T. And since we have cos square T minus sign squat here, that means we could rewrite it as 14 multiplied by the cosine of 2 T. Thus, this is our first derivative. Now we can solve for our second derivative, OK? So we can solve for it by differentiating the first derivative. Now, when we take a good look at our first derivative, 1420, then we can tell we can differentiate it using the chain rule, OK? Because no, by the chain rule, what it basically says is that first we are going to differentiate the cosine of 2 T. OK. So let's write back 14. We're going to differentiate the cosine of 2 T, which is negative sine 2 T. And then we're going to differentiate the inner function, the argument of the cosine, which is 2 T. and the derivative of 2 T is 2. So no, it's gonna be, this is 14 multiplied by negatives to T multiplied by 2, which is gonna be equal to -282 T. Thus, the second derivative of Y is -28 sin to T. If we look back at our answer choices, that tells us that A is the correct answer. Thanks for watching everyone. I hope this video helped.

Find y'' for the following functions.y = cos θ sin θ

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Find y'' for the following functions.
y = cos θ sin θ