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This video on the chain rule, what is the chain rule mean?
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Well, they definitely call it chain for a reason.
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Let's take a look at what circumstances arise or are there that invokes the chain rule.
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So for example, if you look at something like taking the derivative of x squared, for example,
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I want to take the derivative of x squared. I've got some function whose function is
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take the input and square it. Now, what if I want to find the derivative of that function f with respect to x?
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Okay? Well, we all know by now that the power rule says it's 2x.
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But what if that function is just a little bit more complicated? What if I have a new function g of x?
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And it kind of looks like something squared, but that's something that squared is a little more complicated.
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For example, that's something that squared could itself be, for example, a polynomial.
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Say it's 3x squared plus 2x plus 1 squared.
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So in a sense, I know what to do, because if I just think of what's in the parentheses as 1 entity, as 1 input,
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then I should just bring down the 2 and subtract 1, right?
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So if I'm thinking about taking the derivative of this, I think about doing this to the 2 minus 1, for example.
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Now, the question is, what do I put here? And is that the end of the story?
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And the answer is, what you put here is 3x squared plus 2x plus 1. And what you put here, the thing that's missing, is the derivative of that.
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So I'm going to write it like this, 2x plus 1. And I'm going to put a prime on it.
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Scoot over there in a little bit. I'm going to put a prime on it to indicate that I'm going to take the derivative of that.
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Of course, that's easy to take the derivative. I don't need the chain rule for that, because I can take the derivative of the polynomial.
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So how does this work? Well, it turns out that the chain rule, the chain rule, is invoked when you're trying to take the derivative of a composition.
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So what was the composition in this last problem? What was the composition?
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So here I got this function G. It was actually made up of a couple of functions.
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So let's say it was made up of a function H, and H will be a function of X. And that will be this part right here.
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But it was also made up of function that looked like this one, that looked like F, right?
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And to not confuse myself, I'm going to use a different input letter than X, because X is already taken in the function X.
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So in order not to confuse myself, then I'm talking about a different input. I'm going to let F up above be, for example, a function of U, so it's U squared.
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So now if I compose those two functions together, how did I do that symbolically? You already know how to compose two functions symbolically.
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If I want to compose H inside of X, I write that. So the input to F is the function H, right?
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And what is the function H? The function H takes whatever the input is and squares it. So if I was to do this, what would I do? F, I would do what?
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U, well what's U? U is H of X, H of X squared. And if I make the substitution then, I get exactly what I had before.
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2X plus 1 quantity squared.
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So this composition is what I was calling G, and you notice G is a function of the variable of the inside function.
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So this is what I call inside, and the function F is what I call outside.
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So you need to take the derivative of the outside function, evaluated at the inside function. So how does this work?
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Let me take G, D, DX, of G of X is going to be equal to the derivative of F with respect to U, with respect to the outside,
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evaluated at what U is equal to H of X. And then I get a multiply by the derivative of H with respect to X.
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So this is with respect to U, but when I'm all done, I turn it into a function of H by evaluating U at H. And then I got a multiply back by the derivative of H with respect to X.
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So back up here, here's my derivative of H with respect to X. And here is my derivative of F with respect to U. And then I'm evaluating it right here at H of X.
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So I'm doing exactly what these symbols say down here, up above. So as you think of it as one thing, as you practice it is another.
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So what's the easy way to practice these things? I like to use the cross method, not the cross out method, but the cover up method.
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So for example, if I just temporarily put my finger or something over the inside function, I see the outside one is the box raised to the squared.
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So I know what to do. That's two times whatever's in the parentheses to the one power. And now I evaluated it there. That's what this part says.
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So that's in essence leaving what I see there. That's evaluating it at that place. So I just did this part, evaluating it by writing that in there.
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A moment ago, I did this part when I brought the two down and raised it to the one. Now I got a multiply back by the derivative of what's inside.
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And so after I do that, I take the derivative of what was inside the function, the outer function. This case what's inside the box. So if I take that derivative and don't forget, when you take the derivative, you better put parentheses around it or it has multiple terms, you're going to end up being wrong.
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Now that's 6x minus 2. So that's the answer to the problem. Of course we wouldn't write to the first hour here, we just leave it gone. So it's 2 quantity 3x.
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My squared is disappeared on me there. 3x squared minus 2x plus 1 times the quantity 6x minus 2. That's the chain rule. Now the chain rule can be chained. It can be applied over and over again.
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So for example, what happens when you have a function like this? And let's get a new page here.
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What happens when you have something like this? I have this new function, g of x. And we're going to make it 2x plus 1 squared plus x, let's say 2x plus 1. And let's square all that.
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So how are we going to take the derivative of that? Well I'm not going to invoke a bunch of new symbols, but I am going to use quote the cover up method. So the cover up method says you got to find the composition.
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So I'm going to think of that as one entity. So that's that whole guy right there to the two power. So according to the chain rule I take 2 and I evaluate it exactly at what I see. So I get to repeat what I had there.
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So I have 2 plus 1 squared plus 2x plus 1. And then I got to raise that to the first power. Then I got to multiply by, and here's my suggestion to you that you don't try to take the derivative at this point.
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So you just write down what you are going to do. And maybe some brackets might be helpful here. So instead of parentheses, I'm going to put a bracket. So I'm not done until I take the derivative of what's in the brackets.
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So I'm going to get repeated. So do you notice something when I go to take the derivative and brackets, I'm going to have no trouble taking this. The derivative of that is 0, the derivative of that is 2. But this invokes the chain rule again. This invokes the chain rule again, where this is the inner function. So I got to bring down this to raise to the 1 and I got to multiply back by the derivative of the inside.
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Now first of all, in order to be absolutely correct, I got to repeat everything I see up there. So this is kind of a good test of how you're going to organize yourself.
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Because sometimes it's easy if you write it all in the line across, which I didn't do. Now I'm going to multiply back by what? I'm going to multiply back. Let's take the derivative of this piece first. So what is that again? Chain rule 2 times what was inside 2x plus 1.
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Raise to the first power times the derivative of the inside, which is 2. Now I can take the rest of this plus 2 and then the derivative of that is 0. And then like I said earlier, don't forget the parentheses of the incorrect.
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Okay, so there's no at this point nothing more to take the derivative. Yes, it's sloppy. Yes, you could possibly expand and collect. But you have at this point taken the derivative. And for example, the programs which evaluate you're taking the derivative of this would totally accept this answer if you type it in.
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So do you see the chain rule coming here? This whole thing is part of the chain rule of the first inner function, this one. And this right here is the chain rule of the inner inner part right here.
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Okay, so that's the chain rule inside a chain rule. That's why they call it the chain rule, by the way, because you just keep the chaining the derivatives of the inner compositions until you've, quote, undone everything. You've expanded the chain, if you will, of derivatives that are composed inside each other.
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Okay, so that's an example of that. What about when you combine things? What if you have, for example, 2x plus 1 squared times 3x plus 7 cubed.
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Then what you're doing is you're invoking the product rule followed by the chain rule. So the product rule would say 2x plus 1 squared and then do what?
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Take the derivative of that times 3x plus 7 cubed plus what? The derivative or the first one? 2x plus 1 squared times, and then I'll just scrape this over just a little bit, times the derivative of 3x plus 7 cubed.
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Okay, so notice how I can put little brackets around various expressions and put a prime on them to help keep me organized because when you got multiple lines to go here, you don't want to try to jump immediately to the answer. You're just going to end up making a mistake.
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Now I got to keep going. I got to repeat this part here. What's the derivative of this? This is the chain rule applied here, isn't it? The chain rule, bring down the 2, evaluate it at the same place, raise it to the first power, multiply back by the derivative of what was inside, then bring your 3x plus 7 cubed.
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And then over here, you don't do it because it's not derivative, right? But the last parts of the derivative, so you got to do it here, bring down the 3, raise the inside, evaluate it at that same place, raise to the 2, multiply back by the derivative of the inside, which is what?
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And you're all done. Yes, it looks ugly. Yes, you could expand and collect it, but at this point, you have no more derivatives here. And of course, we wouldn't probably write the one right there, and that would probably not be there.
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Okay, so that's the derivative of a product with a chain rule involved as well. All right? That's the chain rule.
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Okay, let's get this out initially.
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Now just a little bit more so you can slide it over and stick it on to it easily, and definitely Europeanrive developers, we're going to see about what is coming of this.