 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 of the chain rule.
You know, people often wonder where the name chain rule comes from. I was just wondering about that myself. So is it because it chains you down? Is it like a chain fence? I decided what it is. It's because by using it, you burst the chains of differentiation, and you can differentiate many more functions using it. So when you want to think of the chain rule, just think of that chain there. It lets you burst free.
Let me give you another application of the chain rule. Ready for this one? So I'd like to differentiate the sin(10t). Again, this is the composite of two functions. What's the inside function? Okay, so I think I'll introduce this new notation. x = 10t, and the outside function is the sine. So y = sin x. So now the chain rule says dy/ dt is... Okay, let's see. I take the derivative of the outside function, and what's that? Sine' and we can substitute because we know what sine' is. So I get cosine of whatever, x, and then times what? Now I differentiate the inside function, which is just 10. So I could write this as 10cos of what? 10t, x = 10t. Now, once you get used to this, this middle variable, you don't have to give a name for it. You can just to think about it in your mind without actually writing it down, d/dt (sin(10t)). I'll just do it again without introducing this middle variable explicitly. Think about it. I first of all differentiate the outside function, and I get cosine. But I don't change the thing that I'm plugging into it. It's still x that I'm plugging into it. x is 10t. So let's just write 10t and not worry about the name of that extra variable. If it confuses you, introduce the new variable. And do it carefully and slowly like this. But, quite quickly, I think you'll get to be able to keep that step in your mind. I'm not quite done yet. I haven't differentiated the inside function, the derivative of 10t = 10. So you get, again, the same result. A little short cut that you'll get used to. Really and truly, once you have the chain rule, the world is yours to conquer. It