U1S4V13 Power Rule.txt

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At the beginning of this sequence,
we calculated the derivative of x squared was 2x.
In this video, we don't want to just talk about x squared.
We want to talk about x to the n for any positive integer n.
So functions like x cubed or x to the sixth,
what would their derivatives be?

Well, let's just do it.
Here's our definition of the derivative.
And for our function x to the n, this
turns out to be limit, as b approaches
x, of b to the n minus x to the n over b minus x.
Now we want to factor this numerator b to the n minus x
to the n like we always do.
And a difference of n-th powers, that factors
as b minus x times b to the n minus 1
plus b to the n minus 2 times x plus b to the n minus 3 times x
squared plus et cetera, et cetera,
until finally we get b times x to the n minus 2
plus x to the n minus 1.
Now if you've never learn this factorization before,
it's actually worth seeing why this works.
It's pretty cool.
So we made a separate video on that
if you want to take a look.
But in any case, our numerator does factor like this.
And then when we divide by b minus x,
this first factor is just going to go away.
So our derivative is just going to be the limit as b approaches
x of this second factor.
So let me just copy and paste it right here.
And this is a polynomial.
So it's continuous.
And to evaluate the limit, we can just plug in x for b.
So our b to the n minus 1 becomes x to the n minus 1.
And then the next term is x to the n
minus 2 times x plus x to the n minus 3 times
x squared et cetera, et cetera.
And then we get x times x to the n minus 2
plus x to the n minus 1.
And notice that all of these terms-- this
is just x to the n minus 1.
This is equal to x to the n minus 1.
They're all equal to x to the n minus 1.
How many x to the n minus 1's do we have?
Well, let's just count the terms up here.
And we're going to look at the exponents on x.
There aren't any x's here, so we can
think of this as x to the 0.
And then we have an x to the 1, an x to the two,
all the way to x to the n minus 1.
So we're going from 0 to n minus 1.
That means that there are n terms,
which means that down here and there's going to be n x
to the n minus 1's.
And when we add them all up, we get n times x to the n minus 1.
And that's our derivative.
This is called the power rule, and it's really handy.
So I'm going to write it up here.
If f of x is x to the n where n is a positive integer, then f
prime of x equals n times x to the n minus 1.

Let's try it out.
I'm going to erase this stuff and let's start
with f of x equals x squared.
So here our exponent n is equal to 2.
So our power rule is going to tell us
that f prime of x equals-- and then we need the n out
in front-- so that's 2.
And we're multiplying it by x to the 2 minus 1.
And that's just 2x.
So that agrees with what we've calculated.
And that's good.
Nothing crazy is going on.
Let's try it out on something new.
Maybe g of x equals x to the sixth.
Here are our exponent is 6, so that's n.
And g prime of x then will be 6--
that goes out in front-- multiplied
by x to the sixth minus 1.
And that equals 6 times x to the fifth.
And voila.

So that's the power rule.
Pretty nifty, isn't it?
Go ahead and play around with it some.
And I'll talk to you in a little bit.