00:01
So let's say we had a profit function, p of x equal to 100 x e to the negative x over 400.
00:11
I'm going to rewrite this as 100 x exp, which means e raised to the negative x over 400.
00:21
This is just so that we can see what we have a little bit better.
00:25
It doesn't really matter which way you write it.
00:27
Either one is fine.
00:28
Let's say that we wanted the change in the profit.
00:37
Let's say we wanted the percent change.
00:43
Okay, well, we have our function for profit.
00:49
So in order to get the change in p, what we're really looking for is the derivative.
00:57
And so the derivative is going to be a product rule because we have this x in here.
01:02
So we're going to have to do the product rule on the p, which means that we'll get 100 exp, or e raised to the negative x over 400.
01:14
So i derive the x.
01:15
Now i'm going to leave the x alone and derive the exponential.
01:20
So that's 100 x.
01:23
And then we're going to have the derivative of e is e.
01:28
So we'll have times the derivative of the inside.
01:31
So we'll have one over, or negative one over 400.
01:37
So that's the derivative of the inside of e times e raised to the negative x over 400.
01:45
All right.
01:48
So this will be the derivative of the, or the change in the profit.
01:52
Now we're not really done yet.
01:57
This is actually, and we'll write this as a dx, or we'll put the x here just to kind of let us know that we took the derivative.
02:07
But what is dx actually? dx is going to be, and so in other words, what i've done here is let's get, let's erase change in profits.
02:20
We'll erase this.
02:21
Yeah.
02:22
We'll erase this.
02:23
We'll erase the equals and just put whole ones for now so that we can say, okay, the change in the profit is approximately equal to dp or the page of the profit, the element infinitesimal change of the profit, right? capital p i'll just put here just so that we can keep up with it.
02:48
Right.
02:49
And so dp is going to be equal to the derivative times dx, right? just so we, we understand while the steps are coming in.
02:57
And so this is actually the change in the profit.
03:00
What we've written here is the change in a profit.
03:04
Okay...