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I just did several videos on the binomial theorem, so I

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think, now that they're done, I think now is good time to do

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the proof of the derivative of the general form.

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Let's take the derivative of x to the n.

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Now that we know the binomial theorem, we

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have the tools to do it.

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How do we take the derivative?

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Well, what's the classic definition of the derivative?

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It is the limit as delta x approaches zero of f of

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x plus delta x, right?

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So f of x plus delta x in this situation is x plus delta

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x to the nth power, right?

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Minus f of x, well f of x here is just x to the n.

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All of that over delta x.

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Now that we know the binomial theorem we can figure out

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what the expansion of x plus delta x is to the nth power.

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And if you don't know the binomial theorem, go to my

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pre-calculus play list and watch the videos on

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the binomial theorem.

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The binomial theorem tells us that this is equal to-- I'm

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going to need some space for this one-- the limit as

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delta x approaches zero.

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And what's the binomial theorem?

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This is going to be equal to-- I'm just going to do the

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numerator-- x to the n plus n choose 1.

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Once again, review the binomial theorem if this is looks like

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latin to you and you don't know latin.

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n choose 1 of x to the n minus 1 delta x plus n choose 2 x to

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the n minus 2, that's x n minus 2, delta x squared.

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Then plus, and we have a bunch of the digits, and in this

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proof we don't have to go through all the digits but the

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binomial theorem tells us what they are and, of course, the

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last digit we just keep adding is going to be 1-- it would

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be n choose n which is 1.

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Let me just write that down. n choose n.

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It's going to be x to the zero times delta x to the n.

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So that's the binomial expansion.

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Let me switch back to minus, green that's x plus delta x

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to the n, so minus x to the n power.

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That's x to the n, I know I squashed it there.

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All of that over delta x.

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Let's see if we can simplify.

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First of all we have an x to the n here, and at the very end

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we subtract out an x to the n, so these two cancel out.

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If we look at every term here, every term in the numerator has

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a delta x, so we can divide the numerator and the

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denominator by delta x.

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This is the same thing as 1 over delta x times

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this whole thing.

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So that is equal to the limit as delta x approaches zero of,

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so we divide the top and the bottom by delta x, or we

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multiply the numerator times 1 over delta x.

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We get n choose 1 x to the n minus 1.

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What's delta x divided by delta x, that's just 1.

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Plus n choose 2, x to the n minus 2.

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This is delta x squared, but we divide by delta x we

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just get a delta x here.

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Delta x.

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And then we keep having a bunch of terms, we're going to divide

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all of them by delta x.

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And then the last term is delta x to the n, but then

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we're going to divide that by delta x.

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So the last term becomes n choose n, x to the zero is 1,

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we can ignore that. delta x to the n divided by delta x.

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Well that's delta x to the n minus 1.

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Then what are we doing now?

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Remember, we're taking the limit as delta

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x approaches zero.

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As delta x approaches zero, pretty much every term that

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has a delta x in it, it becomes zero.

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When you multiply but zero, you get zero.

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This first term has no delta x in it, but

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every other term does.

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Every other term, even after we divided by delta x

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has a delta x in it.

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So that's a zero.

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Every term is zero, all of the other n minus 1

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terms, they're all zeros.

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All we're left with is that this is equal to n choose

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1 of x the n minus 1.

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And what's n choose 1?

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That equals n factorial over 1 factorial divided by n minus 1

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factorial times x to the n minus 1.

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1 factorial is 1.

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If I have 7 factorial divided by 6 factorial, that's just 1.

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Or if I have 3 factorial divided by 2 factorial, that's

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just 3, you can work it out.

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10 factorial divided by 9 factorial that's 10.

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So n factorial divided by n minus 1 factorial,

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that's just equal to n.

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So this is equal to n times x to the n minus 1.

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That's the derivative of x to the n. n times

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x to the n minus 1.

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We just proved the derivative for any positive integer when

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x to the power n, where n is any positive integer.

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And we see later it actually works for all real

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numbers and the exponent.

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I will see you in a future video.

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Another thing I wanted to point out is, you know I said that

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we had to know the binomial theorem.

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But if you think about it, we really didn't even have to know

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the binomial theorem because we knew in any binomial

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expansion-- I mean, you'd have to know a little bit-- but if

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you did a little experimentation you would

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realize that whenever you expand a plus b to the nth

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power, first term is going to be a to the n, and the second

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term is going to be plus n a to the n minus 1 b.

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And then you are going to keep having a bunch of terms.

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But these are the only terms that are relevant to this proof

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because all the other terms get canceled out when delta

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x approaches zero.

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So if you just knew that you could have done this, but it's

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much better to do it with the binomial theorem.

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Ignore what I just said if it confused you.

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I'm just saying that we could have just said the rest of

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these terms all go to zero.

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Anyway, hopefully you found that fulfilling.

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I will see you in future videos.

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