00:02
In this problem, we were asked to show that this function satisfies the laplace laplace equation.
00:08
And the laplace equation is this, that f sub x, x, plus f sub y, y, plus f sub z equals 0.
00:18
So that's what we're trying to show.
00:20
So to take all these second order partial derivatives, we have to start with the first order of partial derivatives.
00:25
Let's begin with f sub x.
00:28
So we see this whole second part, the cosine 5z is just a constant.
00:32
Since we're treating y and z like constants, so that will remain in our answer.
00:36
We need the derivative of this first part.
00:39
So the derivative of e to this whole function up here is itself.
00:45
And then by the chain rule, we need to multiply by the derivative of this exponent with respect to x.
00:52
And at a glance, we can see that the derivative of that exponent with respect to x is just three.
00:58
So expanding that out, let's see what we get.
01:02
We need to put the three in front, really, and, there's not too much else to do.
01:06
We have 3.
01:07
We have that e to the 3x plus 4y, and we had cosine of 5z.
01:14
So that's f sub x.
01:16
Now let's find f sub x.
01:19
So we just take the derivative of this once again.
01:23
The 3 is a constant.
01:24
That cosine 5z is still a constant.
01:27
We just need the derivative once again of e to the 3x plus 4y, and we'll get the same thing as before, e to the 3x plus 4y times 3.
01:36
So we end up with 9e to 3x plus 4y times cosine of 5 z.
01:47
Okay.
01:52
Next up, f suby.
01:55
This will feel much the same.
01:57
We're taking the derivative here with respect to y, that cosine of 5z is a constant.
02:01
The derivative of e to the 3x plus 4y is itself except times.
02:07
Now we take the partial derivative of 3x plus 4y with respect to y.
02:12
Which will give us 4...