 Hello, and welcome to another example of the chain rule. And this one is going to be focusing on trig functions. OK, so the chain rule, again, is used when you have a composition or a composite function, where f is your outside function and g is your inside function. So when you take the derivative of that composite, you end up taking your inside function and plugging into the derivative of the outside, then multiplying that by the derivative of your inside function. So for this example, our function is p of t is sine of 3t squared minus 5. So again, you have to ask yourself the question, which function is being composed into the other function? So what's our outside function and what's our inside function? So our outside, we're going to define to be f. Our inside, we're going to define to be g. So the outside function in this case is the sine of x, t, p, whatever variable you want to use. But I used a t here, so we'll stick with t. So sine of t. The inside is the stuff inside the parentheses. That's what's being plugged into our sine function. So that's going to be 3t squared minus 5. OK, so here we have a trig function. Hopefully you remember how to do the derivative of that one. If not, pause the video, go back and look it up. G is a polynomial. So that one, that should be pretty straightforward. But if you don't remember, again, pause the video, go look it up. So f prime of x, or t in this case, is going to be cosine of t. It's a derivative of sine. And g prime is, this is a power function. So multiply the 3 by the 2 and get a 6. Reduce that power by 1, so 6t minus 5. That piece goes away because it's a constant. OK, so now putting these pieces together for our chain rule, again, it says take our inside function and compose it to plug it into the derivative of that outside function. OK, so that's going to look like cosine of 3t squared minus 5 times the derivative of that inside function, which is 6t. OK. Next example I'd like to look at is y equals cosine squared of theta. OK, this one gets a little bit trickier. So can you identify the inside function, the outside function right away? Maybe you can. But if you're thinking that theta is your inside function, there's really not much to that one. So the derivative of theta is just 1, and you're not really going to get much then out of the chain rule. So this one, if it were me, I would probably do a little bit of algebra first. So I would rewrite this as the cosine of theta quantity squared. Hopefully you guys agree that that's exactly what this cosine squared means. It means you're going to take that cosine function, and then you're going to square it. OK, so hopefully by rewriting it like this, you can now see the outside function and the inside function a little bit easier. OK, so outside, inside. So our f function is, in this case, the outside is theta squared, or x squared, whatever variable you want to use, something squared. The inside function is cosine of theta. OK, what are our derivatives? So f prime, g prime. Again, if you don't remember, pause the video, go back and look it up. But hopefully by now, these are starting to become second nature to you. So the derivative of theta squared is going to be 2 theta. This is a power rule, so you bring the 2 down, reduce your power by 1. So that just brings you down to a 1, which you can put on there, or just leave it as 2 theta. Derivative of cosine theta is negative sine theta. OK, so dy d theta. Again, that notation means the derivative of y, which is what I called this function, with respect to theta, because that's my variable, is, putting these together in our chain rule, we have to take our inside function, which is cosine theta, plug it into the derivative of our outside function. So that's going to look like 2 times cosine theta, and then times, sorry, that's a theta, the derivative of our inside function, which is negative sine theta. Now, I will do a little bit of rewriting here only because I don't like that negative floating in the inside there, in the middle. So I'm going to call this negative 2 cosine theta, sine theta. OK, keep practicing. Thank you for watching.