Name: Proof: d/dx(x^n) | Taking derivatives | Differential Calculus | Khan Academy
Uploaded: 2022-07-12T12:49:40.000Z
Duration: 7 min 3 s
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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.

Now that we know the binomial theorem, we

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

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

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

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

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 tells us that this is equal to-- I'm

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

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

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

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

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

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

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

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

This is the same thing as 1 over delta x times

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.

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

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

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

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

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.

Remember, we're taking the limit as delta

As delta x approaches zero, pretty much every term that

When you multiply but zero, you get zero.

This first term has no delta x in it, but

Every other term, even after we divided by delta x

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

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

That equals n factorial over 1 factorial divided by n minus 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

10 factorial divided by 9 factorial that's 10.

So n factorial divided by n minus 1 factorial,

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

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

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

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

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Proof: d/dx(x^n) | Taking derivatives | Differential Calculus | Khan Academy

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