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Welcome back.

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In the last presentation I showed you that if I had the

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function f of x is equal to x squared, that the derivative of

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this function, which is denoted by f-- look at that, my pen

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is already malfunctioning.

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The derivative of that function, f prime of

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x, is equal to 2x.

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And I used the limit definition of a derivative.

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I used, let me write it down here.

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This pen is horrible.

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I need to really figure out some other tool to use.

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The limit as h approaches 0 -- sometimes you'll see delta x

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instead of h, but it's the same thing-- of f of x plus

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h minus f of x over h.

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And I used this definition of a derivative, which is really

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just the slope at any given point along the curve,

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to figure this out.

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That if f of x is equal to x squared, that

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the derivative is 2x.

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And you could actually use this to do others.

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And I won't do it now, maybe I'll do it in a

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future presentation.

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But it turns out that if you have f of x is equal to x to

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the third, that the derivative is f prime of x is

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equal to 3x squared.

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If f of x is equal to x to the fourth, well then the

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derivative is equal to 4x to the third.

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I think you're starting to see a pattern here.

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If I actually wrote up here that if f of x -- let me see

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if I have space to write it neatly.

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If I wrote f of x -- I hope you can see this -- f

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of x is equal to x.

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Well you know this.

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I mean, y equals x, what's the slope of y equals x?

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That's just 1, right?

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y equals x, that's a slope of 1.

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You didn't need to know calculus to know that.

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f prime of x is just equal to 1.

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And then you can probably guess what the next one is.

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If f of x is equal to x to the fifth, then the derivative is--

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I think you could guess-- 5 x to the fourth.

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So in general, for any expression within a polynomial,

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or any degree x to whatever power-- let's say f of x is

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equal to-- this pen drives me nuts.

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f of x is equal to x to the n, right?

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Where n could be any exponent.

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Then f prime of x is equal to nx to the n minus 1.

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And you see this is what the case was in all

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these situations.

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That 1 didn't show up.

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n minus 1.

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So if n was 25, x to the 25th power, the derivative

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would be 25 x to the 24th.

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So I'm going to use this rule and then I'm going to show

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you a couple of other ones.

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And then now we can figure out the derivative of pretty much

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any polynomial function.

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So just another couple of rules.

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This might be a little intuitive for you, and if you

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use that limit definition of a derivative, you could

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actually prove it.

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But if I want to figure out the derivative of, let's say, the

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derivative of-- So another way of-- this is kind of, what is

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the change with respect to x?

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This is another notation.

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I think this is what Leibniz uses to figure out the

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derivative operator.

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So if I wanted to find the derivative of A f of x, where A

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is just some constant number.

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It could be 5 times f of x.

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This is the same thing as saying A times the

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derivative of f of x.

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And what does that tell us?

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Well, this tells us that, let's say I had f of x.

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f of x is equal to-- and this only works with the constants--

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f of x is equal to 5x squared.

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Right?

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Well this is the same thing as 5 times x squared.

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I know I'm stating the obvious.

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So we can just say that the derivative of this is just 5

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times the derivative of x squared.

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So f prime of x is equal to 5 times, and what's the

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derivative of x squared?

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Right, it's 2x.

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So it equals 10x.

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Right?

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Similarly, let's say I had g of x, just using

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a different letter.

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g of x is equal to-- and my pen keeps malfunctioning.

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g of x is equal to, let's say, 3x to the 12th.

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Then g prime of x, or the derivative of g, is equal

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to 3 times the derivative of x to the 12th.

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Well we know what that is.

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It's 12 x to the 11th.

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Which you would have seen.

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12x to the 11th.

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This equals 36x to the 11th.

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Pretty straightforward, right?

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You just multiply the constant times whatever the

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derivative would have been.

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I think you get that.

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Now one other thing.

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If I wanted to apply the derivative operator-- let me

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change colors just to mix things up a little bit.

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Let's say if I wanted to apply the derivative of operator-- I

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think this is called the addition rule.

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It might be a little bit obvious.

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f of x plus g of x.

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This is the same thing as the derivative of f of x plus

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the derivative3 of g of x.

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That might seem a little complicated to you, but all

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it's saying is that you can find the derivative of each of

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the parts when you're adding up, and then that's the

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derivative of the whole thing.

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I'll do a couple of examples.

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So what does this tell us?

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This is also the same thing, of course.

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This is, I believe, Leibniz's notation.

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And then Lagrange's notation is-- of course these were the

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founding fathers of calculus.

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That's the same thing as f prime of x plus g prime of x.

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And let me apply this, because whenever you apply it, I think

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it starts to seem a lot more obvious.

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So let's say f of x is equal to 3x squared plus 5x plus 3.

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Well, if we just want to figure out the derivative, we say f

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prime of x, we just find the derivative of each

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of these terms.

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Well, this is 3 times the derivative of x squared.

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The derivative of x squared, we already figured

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out, is 2x, right?

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So this becomes 6x.

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Really you just take the 2, multiply it by the 3, and

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then decrement the 2 by 1.

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So it's really 6x to the first, which is the same thing as 6x.

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Plus the derivative of 5x is 5.

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And you know that because if I just had a line that's y equals

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5x, the slope is 5, right?

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Plus, what's the derivative of a constant function?

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What's the derivative of 3?

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Well, I'll give you a hint.

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Graph y equals 3 and tell me what the slope is.

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Right, the derivative of a constant is 0.

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I'll show other times why that might be more intuitive.

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Plus 0.

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You can just ignore that.

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f prime of x is equal to 6x plus 5.

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Let's do some more.

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I think the more examples we do, the better.

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And I want to keep switching notations, so you don't get

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daunted whenever you see it in a different way.

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Let's say y equals 10x to the fifth minus 7x to the

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third plus 4x plus 1.

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So here we're going to apply the derivative operator.

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So we say dy-- this is I think Leibniz's

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notation-- dy over dx.

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And that's just the change in y over the change in x,

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over very small changes.

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That's kind of how I view this d, like a very small delta.

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Is equal to 5 times 10 is 50 x to the fourth minus 21 --

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right, 3 times 7-- x squared plus 4.

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And then the 1, the derivative of 1 is just 0.

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So there it is.

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We figured out the derivative of this very

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complicated function.

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And it was pretty straightforward.

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I think you'll find that derivatives of polynomials are

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actually more straightforward than a lot of concepts that you

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learned a lot earlier in mathematics.

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That's all the time I have now for this presentation.

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In the next couple I'll just do a bunch of more examples, and

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I'll show you some more rules for solving even more

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complicated derivatives.

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See you in the next presentation.

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