 Hello there, and welcome to a screencast where we're going to look at a couple of examples of taking derivatives where the arc sine function is involved. So, first of all, let's remind ourselves of the basic rule about the derivative of arc sine of x, and that is that the derivative with respect to x of arc sine x is 1 over the square root of 1 minus x squared. It's kind of a strange looking rule. You might not expect that to be the derivative of an inverse trig function. But the derivation we've seen before, and you can go back and review that. We need to also just mention that this is going to hold as long as x is between negative one and positive one because of the situation in the square root here. So with that basic rule, we can go and compute some more complicated derivatives. Let's start with y equals arc sine of x divided by radical x. So how am I going to approach this? Well, like we've seen before in deciding which rule I should use first in a given problem, I want to zoom out and think about how the function is put together on a global scale. And what I see here is that this function is basically of the form something divided by something else. Okay, I have a function on the top and a function on the bottom. What I'm really doing here is dividing two things. And so when I think about the derivative rule I'm going to use, I need to use the quotient rule because this is not a product, it's not a composite, it's a quotient. So I'm going to use the quotient rule to find my derivatives. Let's walk through that. So y prime would be the bottom function times the derivative of the top function. I'll just write out that rule here, the derivative of arc sine of x. There we go. Minus the derivative of the bottom function times the top function. And this large fraction here is over the bottom function squared. There we go. So there's the basic setup for the quotient rule. I'm just going to do one more step here. There's a lot of algebra that can be simplified here, but just to keep this from getting too boring. I'm just going to kind of do the calculus here and leave the algebra for another day. And that's just mainly I have two derivatives to take here. So y prime is square root of x. And now I'm going to use my basic formula here for the derivative of arc sine. And that's one divided by radical one minus x squared. And then I have a minus, the derivative of one of a square root of x. Square root of x is the same thing as x to the one-half. And so its derivative is one-half x to the minus one-half. And this is all times arc sine of x. And then on the bottom I have the square root of x squared and that just comes out to x. So some simplification can take place here, but I don't want to obscure the calculus that's taken place. The main idea here is that I'm using the quotient rule based on the structure of the formula. Now let's look at another example here, where I'm looking at the derivative of arc sine of natural log of x. So again, I want to decide which rule I'm going to apply here to take the derivative. So what kind of function is this? Pretty clearly not a quotient this time. Is it a product? Because I see two things kind of juxtaposed together, arc sine and log x. Now this is not a product, this is a composite function. I have one function stuffed inside another. Remember when I write this right here, arc sine of natural log x. It's not multiplication. That's not something called arc sine times natural log of x. That's arc sine of natural log of x. And so I have a composite function in front of me. And it appears that the inside function, which I usually call g of x, is the natural log function. And the outside function is what we often call f and that is arc sine of x. So now that I see it's a composite function, if I took this inner function and loaded it in right here, I would have my original y. I know which rule to use and that's the chain rule. So let's go and set up the chain rule over here and see how far we can go. So the chain rule would say the derivative of y is f prime of g of x times g prime of x. Now let's just walk through these derivatives here one at a time. f is the arc sine function. It's derivative is 1 over radical 1 minus. I'm going to put a blank here squared. In the formula, in the basic rule, it says x right there. But I'm going to put something in for x. And so like we've done in previous chain rule screencasts, we're going to leave a blank there. What exactly do we put in that blank? Well, we put in g of x, that's right here. That's the natural log function. So that gets kind of tiny, but it's 1 over radical 1 minus natural log of x squared. And then I have to end this off by multiplying by the derivative of g. g again is the natural log function and so that is 1 over x for its derivative. It's not really much we can do to simplify that other than to just sort of put the fractions together by multiplying. So the final result here is 1 over x times 1 minus natural log x squared. Again, no real surprises here as long as we analyze the structure properly of the function and know what kind of function it is. Composite, product, quotient, whatever, that tells us what rule to use. So thanks for watching.