00:01
So this problem wants you to calculate the gradient of the function x squared plus y square plus z squared to the negative one half power plus ellen of xyz at the point negative one to negative two.
00:15
All right.
00:16
So if we want to calculate, well, well if you want to calculate the gradient of a function, it will be a vector comprised of its partial derivatives with respect to x, respect to y, and with respect to z.
00:31
So let's go ahead and calculate the partial derivatives of this very long function.
00:38
So we want to take the partial derivative with respect to x, you would have to first differentiate the first term.
00:46
And we have to use chain rule.
00:48
So we would get negative 1 half times x squared plus y squared plus z squared to the negative 3 half power because of power rule times 2x, because of chain rule, we have to differentiate this part afterwards.
01:07
Then we have to differentiate the natural logarithm, which is just 1 over x, y, z.
01:15
Then we differentiate x, y, z with respect to x, and that just gives us, that just gives us y z in the numerator.
01:24
And you can notice that this, and this will cancel out, leaving us with just 1x.
01:30
So that's our, this is the partial derivative with respect to x.
01:36
Now let's take the partial derivative with respect to y...