Learn about Calculus: Derivatives 5 Video

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Learn about Calculus: Derivatives 5

I’m not going to do a bunch more examples using the chain rule, so let's see. Once again, if I had f(x) = (x3 + 2x2 – x-2), we haven’t put any negative exponents yet. But I think you’ll see that at the same patterns apply and all of that to let's say the -7. We want to figure out what f prime of x, what the derivative of f(x) is. So this might seem very complicated and daunting to you and obviously to take this entire polynomial to the -7 power would take you forever. But using the chain rule, you can do it quite quickly. So the first thing we want to do is we want to take the derivative of the inner function, I guess we could call it. We want to take it the derivative of this and once the derivative of x3 + 2x2 - x-2. Well we know how to do that, that was the first kind of type of derivatives we learned how to do. It's 3x2 and then 3x2 + 4x1 or just for 4x and then here with the negative derivative, we do the exact—on negative exponent we do the exact same thing. We say -2 x -1, there's a 1 here, but we don’t write it down. So -2 x -1 is + 2x to the—and then we decrease the exponent by 1, so its x-3 right? So we figure out what the derivative of the inside is and then we just multiply that whole thing times the derivative of kind of the entire expression. So then that will be—we take the -7, we’ll do different colors. So this is the entire thing then we take -7, so its times -7, this whole expression I’m going to run out this phase (x3 +2x2 – x-2)—that’s –x-2 and all of that we just decrease this exponent by 1-8. So when we write it all down a little bit neater now. So we get f1(x) so the derivative of f1(x) = (3x2 + 4x + 2x-3), so I don’t know why I did that, that’s -3, x -7 (x3 + 2x2 – x-2) all of that to the -8 power. And could simplify it a little bit, maybe we could just multiply this at -7 times you could distribute it across these expressions. So we’d say that equals -7 so that it's equal (-21x2 – 28x -14x-3) (x3 + 2x2 – x-2)-8. So there we did it. We took this, what I would say is a very complicated function and using the chain rule and just some with the basic rules that I had introduce to a couple of presentation though we we’re able to find the derivative of it. And now if we wanted to for whatever application we could find the slope of this function at any point x by just substituting that point into this equation and look at the slope of that point. So let me do a slightly harder one to show that the chain rule, you could have it go or arbitrary deepen the chain rule. Once again, I don’t know—let me see, clear the image. Oh that was pretty fast, look at that. Okay, so let's say I had—let me see if I can write a little bit thinner. See if I had f(x)—I don’t know if you can see that so I’m going to do it a little bit fatter. F(x) is equal to—I want to make it a little bit more complicated this time. (3x-2 + (5x3 -7x)5 and then this whole expression to the 3rd power. So as you're saying south, you’re starting to go nuts, this is going to take us forever. Well I’ll show you using the chain rule; it will not take that long. So the way I think about it, so f1(x) =, I start of kind of what the innermost functions. So let me see if I can use colors to make it a little bit simpler. So let's take the derivative of this innermost function first, right? I mean—let me actually give you the big picture. We want to find the derivative of the innermost function and then a little bit bigger and then a little bit more big than that. I know that’s not precise mathematical terms but you’ll get the point when see and I show you this example. So first we’ll do this inner function, this inner expression and the derivative of that’s pretty easy right? Its (15x2 – 7) right, that was pretty straight forward. And now we’re going to want to multiply that times this entire derivative here. So let me circle that on different. So then we want to do this. We’re going to multiply it that times this entire derivatives. Well that’s just times 5 and we just pretend like this was just x here right because the derivative of x5 is 5x4 right? But instead of an x, we have this whole expression, (5x3 - 7x)—we’ll write that. (5x3 – 7x) and now the exponent here goes down by 1. So its 5 (5x3 – 7x)4 so we figure out the derivative of this so far and then we want to figure out the derivative of this so we’ll add it because we’re trying to figure out the derivative of this entire expression so this is an easy one. Let me draw that in a different color. So then we want the derivative of this. So that’s (-2 x 3) so that’s -6x-3. So what have we done so far? We’ve so far figure out the derivative of this entire expression, right? The derivative of that entire expression using the chain rule is this. And now we’re almost done, we just have to multiply that so I’m just going to just—I run out of space on that line but let's just assumed that the line continues. So that’s times—and now we just take the derivative of kind of these whole big things. And now it's going to be the derivative of—I’m going to use this brown color. So it's whole big expression to the 3rd power right. So that becomes .3 (3x + (5x – 7x)5)2. That was an ultra confusing example and this is probably the hardest chain rule problem you’ll see in a lot of the questions you’ll have on your test. But you see it wasn’t that difficult. We just kind of went to the smallest possible function and actually the small function will would have been one of these terms but we just found the derivative of this which was (15x2 – 7) and then we just used the principle that the derivative of kind of a function is this the derivative of kind of a function is the derivative of each of its parts. Well actually, the derivative—we figure out the derivative of this inner piece which was (15x2 – 7) and then we multiplied it times the derivative of this slightly larger piece which is 5 times this entire expression to the 4th. Then we added to the derivative of 3x then the -2 and then that whole thing—and actually I should put a big parenthesis around here. That whole thing, we multiplied it times the derivative of this larger expression. I think I might have confused you, so I apologize if I have and then on the next presentation I’m just going to do a bunch more of chain rule problems and at some point, it should start to makes sense to you. I think it's just a matter of seeing example after example after example. I’ll see into the next presentation and I apologize if I had confused you.

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I’m not going to do a bunch more examples using the chain rule, so let's see. Once again, if I had f(x) = (x3 + 2x2 – x-2), we haven’t put any negative exponents yet. But I think you’ll see that at the same patterns apply and all of that to let's say the -7. We want to figure out what f prime of x, what the derivative of f(x) is. So this might seem very complicated and daunting to you and obviously to take this entire polynomial to the -7 power would take you forever. But using the chain rule, you can do it quite quickly.

So the first thing we want to do is we want to take the derivative of the inner function, I guess we could call it. We want to take it the derivative of this and once the derivative of x3 + 2x2 - x-2. Well we know how to do that, that was the first kind of type of derivatives we learned how to do. It's 3x2 and then 3x2 + 4x1 or just for 4x and then here with the negative derivative, we do the exact—on negative exponent we do the exact same thing. We say -2 x -1, there's a 1 here, but we don’t write it down.

So -2 x -1 is + 2x to the—and then we decrease the exponent by 1, so its x-3 right? So we figure out what the derivative of the inside is and then we just multiply that whole thing times the derivative of kind of the entire expression. So then that will be—we take the -7, we’ll do different colors. So this is the entire thing then we take -7, so its times -7, this whole expression I’m going to run out this phase (x3 +2x2 – x-2)—that’s –x-2 and all of that we just decrease this exponent by 1-8. So when we write it all down a little bit neater now. So we get f1(x) so the derivative of f1(x) = (3x2 + 4x + 2x-3), so I don’t know why I did that, that’s -3, x -7 (x3 + 2x2 – x-2) all of that to the -8 power. And could simplify it a little bit, maybe we could just multiply this at -7 times you could distribute it across these expressions.

So we’d say that equals -7 so that it's equal (-21x2 – 28x -14x-3) (x3 + 2x2 – x-2)-8. So there we did it. We took this, what I would say is a very complicated function and using the chain rule and just some with the basic rules that I had introduce to a couple of presentation though we we’re able to find the derivative of it. And now if we wanted to for whatever application we could find the slope of this function at any point x by just substituting that point into this equation and look at the slope of that point.

So let me do a slightly harder one to show that the chain rule, you could have it go or arbitrary deepen the chain rule. Once again, I don’t know—let me see, clear the image. Oh that was pretty fast, look at that.

Okay, so let's say I had—let me see if I can write a little bit thinner. See if I had f(x)—I don’t know if you can see that so I’m going to do it a little bit fatter. F(x) is equal to—I want to make it a little bit more complicated this time. (3x-2 + (5x3 -7x)5 and then this whole expression to the 3rd power. So as you're saying south, you’re starting to go nuts, this is going to take us forever. Well I’ll show you using the chain rule; it will not take that long.

So the way I think about it, so f1(x) =, I start of kind of what the innermost functions. So let me see if I can use colors to make it a little bit simpler. So let's take the derivative of this innermost function first, right? I mean—let me actually give you the big picture. We want to find the derivative of the innermost function and then a little bit bigger and then a little bit more big than that. I know that’s not precise mathematical terms but you’ll get the point when see and I show you this example.

So first we’ll do this inner function, this inner expression and the derivative of that’s pretty easy right? Its (15x2 – 7) right, that was pretty straight forward. And now we’re going to want to multiply that times this entire derivative here. So let me circle that on different. So then we want to do this. We’re going to multiply it that times this entire derivatives. Well that’s just times 5 and we just pretend like this was just x here right because the derivative of x5 is 5x4 right? But instead of an x, we have this whole expression, (5x3 - 7x)—we’ll write that.

(5x3 – 7x) and now the exponent here goes down by 1. So its 5 (5x3 – 7x)4 so we figure out the derivative of this so far and then we want to figure out the derivative of this so we’ll add it because we’re trying to figure out the derivative of this entire expression so this is an easy one. Let me draw that in a different color. So then we want the derivative of this. So that’s (-2 x 3) so that’s -6x-3.

So what have we done so far? We’ve so far figure out the derivative of this entire expression, right? The derivative of that entire expression using the chain rule is this. And now we’re almost done, we just have to multiply that so I’m just going to just—I run out of space on that line but let's just assumed that the line continues. So that’s times—and now we just take the derivative of kind of these whole big things.

And now it's going to be the derivative of—I’m going to use this brown color. So it's whole big expression to the 3rd power right. So that becomes .3 (3x + (5x – 7x)5)2. That was an ultra confusing example and this is probably the hardest chain rule problem you’ll see in a lot of the questions you’ll have on your test. But you see it wasn’t that difficult. We just kind of went to the smallest possible function and actually the small function will would have been one of these terms but we just found the derivative of this which was (15x2 – 7) and then we just used the principle that the derivative of kind of a function is this the derivative of kind of a function is the derivative of each of its parts.

Well actually, the derivative—we figure out the derivative of this inner piece which was (15x2 – 7) and then we multiplied it times the derivative of this slightly larger piece which is 5 times this entire expression to the 4th. Then we added to the derivative of 3x then the -2 and then that whole thing—and actually I should put a big parenthesis around here. That whole thing, we multiplied it times the derivative of this larger expression.

I think I might have confused you, so I apologize if I have and then on the next presentation I’m just going to do a bunch more of chain rule problems and at some point, it should start to makes sense to you. I think it's just a matter of seeing example after example after example.

I’ll see into the next presentation and I apologize if I had confused you.

Transcription by: Scribe4you Transcription Services