00:01
Hi there, so for this problem we are given the function f that depends on x, y, and z that is equal to x elevated to y divided by z.
00:13
Now we need to find the value of the following partial derivatives.
00:17
So for part a of this problem, we need to find the partial derivative with respect to x, and then evaluate that at 3, 4, 4.
00:29
Right so then when we do the partial derivative of the function that we are given with respect to x this is the variable at which we are differentiated so we can associate this and if we will have something for example x to the a so we know that its derivative is just equal to we take down a and then we subtract one exponent to to the current exponent so in this case, that will be y divided by z times x elevated to y divided by z minus 1.
01:12
Okay? and now we just need to evaluate this at the values that we are given.
01:18
This means that the value of x is 3, the value of y is 4, and the value of z is 4.
01:24
So that will be 4 divided by 4 times 3 elevated to 4 divided by 4, which is 1 minus 1.
01:36
That will give us 0.
01:38
And something elevated to 0 is 1, so this will give us just simply 1.
01:42
So that will be the solution for the first partial derivative.
01:45
Now for the second partial derivative, derivative, we are asked about the partial derivative with respect to y of 4, 4, 2.
02:00
Now, in this case, what we need to do is, well, now the variable is y.
02:10
It's very simple if we will have something like let's say a constant a because now x and z behaves like like constants so we will have then y divided by let's say another number b so we know that if we differentiate this the derivative of this is first we will have the neperian logarithm of a because because that's what we will obtain if we will have a base that is different to the lr number.
02:48
And then this times a elevated to y divided by b, and then this times the number that multiplies y, which in this case is the inverse of b.
03:00
So that will be the solution...