ues of n. So there's something new in your list of rules that you can apply, of values of the derivative. That standard rule is true for negative as well as positive exponents. And that comes out of a quotient rule.
Okay, so we've done two rules. I've talked about the product rule and the quotient rule. What's next? Let's see the chain rule. So this is a composition rule. So the kind of thing that I have in mind, composition of functions is about substitution. So the kind of function that I have in mind is, for instance, y = (sin t)^10. That's a new one. We haven't seen how to differentiate that before, I think. This kind of power of a trig function happens very often. You've seen them happen, as well, I'm sure, already. And there's a little notational switch that people use. They'll write sin^10(t). But remember that when you write sin^10(t), what you mean is take the sine of t, and then take the tenth power of that. It's the meaning of sin^10(t). So the method of dealing with this kind of composition of functions is to use new variable names. What I mean is, I can think of this (sin t)^10.
I can think of it it as a two step process. First of all, I compute the sine of t. And let's call the result x. There's the new variable name. And then, I express y in terms of x. So y says take this and raise it to the tenth power. In other words, y = x^10. And then you plug x = sine of t into that, and you get the formula for what y is in terms of t. So it's good practice to introduce new letters when they're convenient, and this is one place where it's very convenient.
So let's find a rule for differentiating a composition, a function that can be expressed by doing one function and then applying another function. And here's the rule. Well, maybe I'll actually derive this rule first, and then you'll see what it is. In fact, the rule is very simple to derive. So this is a proof first, and then we'll write down the rule. I'm interested in delta y / delta t. y is a function of x. x is a function of t. And I'm interested in how y changes with respect to t, with respect to the original variable t. Well, because of that intermediate variable, I can write this as ((delta y / delta x) (delta x/ delta t)). It cancels, right? The delta x cancels. The change in that immediate variable cancels out. This is just basic algebra. But what happens when I let delta t get small? Well this give me dy /dt. On the right hand side, I get (dy/dx) (dx/dt). So students will often remember this rule. This is the rule, by saying that you can cancel out for the dx's. And that's not so far from the truth. That's a good way to think of it.
In other words, this is the so-called chain rule. And it says that differentiation of a composition is a product. It's just the product of the two derivatives. So that's how you differentiate a composite of two functions. And let's just do an example. Let's do this example. Let's see how that comes out. So let's differentiate, what did I say? (sin t)^10. Okay, there's an inside function and an outside function. The inside function is x as a function of t. This is the inside function, and this is the outside function. So the rule says, first of all let's differentiate the outside function. Take dy/ dx. Differentiate it with respect to that variable x. The outside function is the 10th power. What's it's derivative? So I get 10x^9. In this account, I'm using this newly introduced variable named x. So the derivative of the outside function is 10x^9. And then here's the inside function, and the next thing I want to do is differentiate it. So what's dx /dt, d/dt (sine t), the derivative of sine t? All right, that's cosine t. That's what the chain gives you. This is correct, but since we were the ones to introduce this notation x here, that wasn't given to us in the original problem here. The last step in this process should be to put back, to substitute back in what x is in terms of t. So x = sin t. So that tells me that I get 10(sin(t))^9, that's x^9, times the cos(t). Or the same thing is sin^9(t)cos(t). So there's an application 