00:01
So we're looking to find the first partial derivatives of this function.
00:04
So let's start by finding ds dy.
00:07
Okay, so the way to do this is just to pretend that z is a constant, so just pretend it's any constant, you know, think of it as k or whatever makes it easier for you, and instead just differentiate this with respect to y.
00:21
So we can see that the only y is inside the tan function.
00:25
So tan normally differentiates, let's say tan theta, that differentiates to sec squared theta.
00:32
But what about tan of k theta? well that differentiates to k sec squared theta.
00:39
Okay, so in this case the 3z cubed stays where it is, and then tan yz is going to differentiate to z sec squared yz, because remember that z, we're just treating it as a constant.
00:54
So our answer is 3z to the 4 sec squared yz.
01:03
Okay, so that's ds dy, now let's find ds dz.
01:07
So we're gonna have to use the product rule on this.
01:09
So first we differentiate 3z cubed, and that becomes 9z squared, and we need to multiply that by tan of yz...