00:03
Given that y is equal to x to the 2x, we want to find the derivative of y with respect to x or d y d x.
00:10
In order to do this, we can't use a regular old power rule, even though it's x to a power.
00:13
The power rule works when we have constant powers.
00:18
This is a variable power.
00:19
It's raised to the 2x.
00:20
So we're going to have to do something called logarithmic differentiation.
00:24
So what i'm going to do is i'm going to take the natural log of both sides.
00:27
So i make it the natural log of y is equal to the natural log of x raised to the 2x.
00:33
So i still have an equation because i did the same thing to both sides.
00:36
And the reason that we do that, i'm going to leave this left side alone, but natural logs have some nice properties.
00:42
If we have the natural log of x to a power, we can bring that power and make it a multiple instead.
00:47
So the natural log of x, the 2x, is the same as 2x times the natural log of x.
00:55
And from there we can take its derivative.
00:58
I'll do that in green here.
00:59
So the derivative of the natural log of y, well, that is.
01:03
D -y -d -x or y -prime, whichever way you want to write that over itself.
01:07
It's the derivative of the function over itself is equal to.
01:10
And now i have to take the derivative of this side...