U1S4V15 Extended Power Rule.txt

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You've been playing around with the power rule.
And hopefully you're getting the sense that it's really useful.
In this video, we're going to see that it's even more useful
than you originally thought.
So here's the power rule.
It tells us the derivative of functions.
F of x equals x to the n.
So our input variable x is being raised to some power n.
And we've been using it so far in cases
where n is a positive integer.
So our function might have been x squared or x
to the sixth, those sorts of things.
And if you think back to how we proved the power rule,
that's really appropriate.
Because we proved it using polynomials and factorization.
And so we were assuming that n was a positive integer back
then.
The exciting news is that the power rule is actually true
if n is any number-- any fixed number whatsoever.
We're not going to prove this just yet.
We'll save that for later sections.
But we do want you to know this fact
and be able to use it right now.
OK?

The first example that we'll do is f of x equals 1 over x.
I know we've done this one before,
but let's just see how the general power rule applies.
This function 1 over x, it can be written as a power.
It's x to the minus 1.
So this fits perfectly into the general power rule.
N here is minus 1.
And so f prime of x is just going to be--
and we need to bring the n out in front.
So minus 1 times x to the minus 1 minus 1.
And so that's just minus 1 times x to the minus 2.
Or if you want that without a negative exponent,
we've got minus 1 over x squared.
And that's our derivative.
It's the same one that we got before.
So that's good.
We're not doing anything crazy here.
Let's do another example, one that we haven't seen before.
How about g of t equals the square root of t.
So I've switched here the input variable to t rather than x.
But it's the same principle.
In order to use this power rule, we should be asking ourselves
is this function a power of t?

Indeed, it is.
The square root of t is the same thing as t to the 1/2.
So it fits.
And thus to take the derivative g prime of t,
we're going to want the exponent, 1/2, out in front.
And then we're going to have t to a power.
It's going to be t to the 1/2 minus 1.
And that equals 1/2 t to the minus 1/2,
or 1/2 times 1 over the square root of t,
if you like that better.

So now you might be saying, oh my gosh,
this general power rule is so great.
Is there anything it can't do?
Well, we do have to give you a warning here,
which is that the power rule only works when the exponent n
is a fixed number and the base is the input variable
for your function -- so in this case x,
since we wrote our function as f of x.
So for instance, it does not apply to a function like 2
to the x.
2 to the x, our exponent is x.
That's not fixed.
And our base is 2.
That's not our variable.
So it doesn't work.
It does not apply to something like g of t
equals cosine of t cubed.
Here, yes, it's cubed.
So that's a fixed power.
But the base is not just t.
It's some function of t.
So the power rule doesn't apply directly to that.
If we have something like h of x equals x to the x,
now our base is x.
That's good.
That's our input variable.
But our exponent is not fixed.
It's x, so it's moving around.
So with all of these, the power rule
won't work, at least not by itself.
We'll learn how to differentiate all of them eventually.
But for now, we just don't have enough tools.
OK?

We've got some questions for you to play around
with this general power rule.
And hopefully you can figure out just where it's useful
and where it's not.