 Let's take a look at an example of how to use the chain rule in order to help us find us the location of horizontal tangents to a curve. Now remember that if we're going to have a horizontal tangent to a curve, that means that the derivative there is going to equal zero. So what we're going to have to do is find the derivative of this given function, set it equal to zero and solve it. So derivative of sine is cosine, so we'd start out having two cosine of x. Now for the term after the plus sign, sine squared of x, that's sine of x, the quantity squared. That's the chain rule part, alright, so you have to think through how to do the chain rule. So if we started on the outside, we'd have two sine of x, but then we have to multiply by the derivative of sine, which is going to be cosine. And this is what we're setting equal to zero. Now it becomes a trig equation to solve. Notice that we do have a greatest common factor in cosine, so let's factor that out. Actually we can pull the two as well, so we have two cosine of x as our greatest common factor that is multiplied by one plus sine of x. And of course you can use the distributive property to check to make sure that is correct. So according to the zero product property, we have two cosine of x equals zero. So we'll solve that in just a second. We have one plus sine of x equals zero and we shall solve that as well. So over on the left, of course if we divide out the two, we're looking at cosine of x equaling zero. Now think of your unit circle and think about where cosine equals zero. That's going to happen at the top of the circle at pi over two and at the bottom of the circle at three pi over two. We are of course assuming in this problem we're just talking about angles from zero to two pi. I guess we should have stated that from the beginning. Because of course there are going to be infinitely many angles that have a cosine of zero. But if we just think of one complete circle from zero to two pi, we would have pi over two and three pi over two. Now on the other side, on the right, we need to think about where it is that sine of x is equal to negative one. Well that's happening at the bottom of the circle as well at three pi over two. So three pi over two does show up twice as an answer. So here we have two locations in between zero and two pi on this particular curve at which we would have horizontal tangents. So of course you can graph it on your graphing calculator to confirm it and check it graphically to make sure what you did algebraically is correct.