Power rule
Proof: d/dx(x^n) Proof that d/dx(x^n) = n*x^(n-1)
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- I just did several videos on the binomial theorem, so I
- think, now that they're done, I think now is good time to do
- the proof of the derivative of the general form.
- Let's take the derivative of x to the n.
- Now that we know the binomial theorem, we
- have the tools to do it.
- How do we take the derivative?
- Well, what's the classic definition of the derivative?
- It is the limit as delta x approaches zero of f of
- x plus delta x, right?
- So f of x plus delta x in this situation is x plus delta
- x to the nth power, right?
- Minus f of x, well f of x here is just x to the n.
- All of that over delta x.
- Now that we know the binomial theorem we can figure out
- what the expansion of x plus delta x is to the nth power.
- And if you don't know the binomial theorem, go to my
- pre-calculus play list and watch the videos on
- the binomial theorem.
- The binomial theorem tells us that this is equal to-- I'm
- going to need some space for this one-- the limit as
- delta x approaches zero.
- And what's the binomial theorem?
- This is going to be equal to-- I'm just going to do the
- numerator-- x to the n plus n choose 1.
- Once again, review the binomial theorem if this is looks like
- latin to you and you don't know latin.
- n choose 1 of x to the n minus 1 delta x plus n choose 2 x to
- the n minus 2, that's x n minus 2, delta x squared.
- Then plus, and we have a bunch of the digits, and in this
- proof we don't have to go through all the digits but the
- binomial theorem tells us what they are and, of course, the
- last digit we just keep adding is going to be 1-- it would
- be n choose n which is 1.
- Let me just write that down. n choose n.
- It's going to be x to the zero times delta x to the n.
- So that's the binomial expansion.
- Let me switch back to minus, green that's x plus delta x
- to the n, so minus x to the n power.
- That's x to the n, I know I squashed it there.
- All of that over delta x.
- Let's see if we can simplify.
- First of all we have an x to the n here, and at the very end
- we subtract out an x to the n, so these two cancel out.
- If we look at every term here, every term in the numerator has
- a delta x, so we can divide the numerator and the
- denominator by delta x.
- This is the same thing as 1 over delta x times
- this whole thing.
- So that is equal to the limit as delta x approaches zero of,
- so we divide the top and the bottom by delta x, or we
- multiply the numerator times 1 over delta x.
- We get n choose 1 x to the n minus 1.
- What's delta x divided by delta x, that's just 1.
- Plus n choose 2, x to the n minus 2.
- This is delta x squared, but we divide by delta x we
- just get a delta x here.
- Delta x.
- And then we keep having a bunch of terms, we're going to divide
- all of them by delta x.
- And then the last term is delta x to the n, but then
- we're going to divide that by delta x.
- So the last term becomes n choose n, x to the zero is 1,
- we can ignore that. delta x to the n divided by delta x.
- Well that's delta x to the n minus 1.
- Then what are we doing now?
- Remember, we're taking the limit as delta
- x approaches zero.
- As delta x approaches zero, pretty much every term that
- has a delta x in it, it becomes zero.
- When you multiply but zero, you get zero.
- This first term has no delta x in it, but
- every other term does.
- Every other term, even after we divided by delta x
- has a delta x in it.
- So that's a zero.
- Every term is zero, all of the other n minus 1
- terms, they're all zeros.
- All we're left with is that this is equal to n choose
- 1 of x the n minus 1.
- And what's n choose 1?
- That equals n factorial over 1 factorial divided by n minus 1
- factorial times x to the n minus 1.
- 1 factorial is 1.
- If I have 7 factorial divided by 6 factorial, that's just 1.
- Or if I have 3 factorial divided by 2 factorial, that's
- just 3, you can work it out.
- 10 factorial divided by 9 factorial that's 10.
- So n factorial divided by n minus 1 factorial,
- that's just equal to n.
- So this is equal to n times x to the n minus 1.
- That's the derivative of x to the n. n times
- x to the n minus 1.
- We just proved the derivative for any positive integer when
- x to the power n, where n is any positive integer.
- And we see later it actually works for all real
- numbers and the exponent.
- I will see you in a future video.
- Another thing I wanted to point out is, you know I said that
- we had to know the binomial theorem.
- But if you think about it, we really didn't even have to know
- the binomial theorem because we knew in any binomial
- expansion-- I mean, you'd have to know a little bit-- but if
- you did a little experimentation you would
- realize that whenever you expand a plus b to the nth
- power, first term is going to be a to the n, and the second
- term is going to be plus n a to the n minus 1 b.
- And then you are going to keep having a bunch of terms.
- But these are the only terms that are relevant to this proof
- because all the other terms get canceled out when delta
- x approaches zero.
- So if you just knew that you could have done this, but it's
- much better to do it with the binomial theorem.
- Ignore what I just said if it confused you.
- I'm just saying that we could have just said the rest of
- these terms all go to zero.
- Anyway, hopefully you found that fulfilling.
- I will see you in future videos.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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