00:01
So for this question we're going to be taking a bunch of partial derivatives.
00:05
There are quite a few questions that we're going to get through, so i will be doing them relatively fast.
00:10
So with this question we have our function, we want an x partial, so the derivative of natural log is 1 over the input times whatever the derivative is on the inside.
00:31
Derivative of x is 1 plus the derivative of a square root is 1 over 2 times the root, and then once more chain rule just for this piece, and the derivative of the inside with respect to x is 2x.
00:54
So there's the partial derivative, now we want to evaluate when x is 3 and y is 4.
01:09
Okay so let's see, the x squared plus y squared in this case will be 25, right it's a 3 4 5 triangle, so the square root in total is 5.
01:22
So i have 1 over x plus 5, 1 over x plus 5, 1 plus 2 times 5, 2 and then 3.
01:46
Okay let's simplify this down, notice the 2's cancel, so inside parentheses here we have 1 plus 3 fifths, so that's, what is that, 8 fifths, and then you also have this 1 over 8, and once again you get some more canceling for a final answer of 1 fifth.
02:20
Okay second part we have a function of three variables x, y, and z.
02:32
It is x plus y plus z in the denominator with a y in the numerator.
02:43
Our goal is to find the y partial evaluated at 2, 1, and negative 1.
02:53
Okay so let's find our y partial, because we have y in the numerator and denominator i'm using quotient rule, so bottom function derivative of the top minus top function derivative of the bottom.
03:10
The derivative of x and z are both 0, so that entire bottom derivative is 1, and this is over the bottom function squared.
03:25
We can simplify this a little bit because we have x plus y plus z minus y, so in reality it's just x plus z, and then over that denominator squared.
03:43
So then we can calculate the partial evaluated at 2, 1, negative 1.
03:51
So let's see, in the denominator we add all three components together, which is 2, square it, you get 4.
03:59
X plus z, 2 minus 1, there we go.
04:04
Alright so that's it for the first question.
04:07
Now the second question wants us to calculate all second partial derivatives.
04:12
So there are going to be again quite a few that end up coming out.
04:15
So for the first part we have that our function is equal to, let's see we have x cubed y to the fifth plus 2 x to the fourth times y.
04:35
Alright let's go ahead and do, first we need the x partial, derivative of x cubed is 3 x squared, the y's are constant, derivative of x to the fourth is 4 x cubed, and there we go.
04:55
Next we can get our double x partial, okay derivative of x squared is 2 x, okay derivative of x cubed 3 x squared.
05:10
Okay so there's one of the second partials.
05:14
Okay now going back up to fx we can also find the mixed partial.
05:20
So now taking the y partial, let's see derivative of y to the fifth is 5 y to the fourth, so i have 15 x squared y to the fourth, the derivative of y is 1, leaving us just with the 8 x cubed.
05:38
Okay so there's the mixed partial.
05:43
Okay what else do we need? we need the y partial, so going back to the original function the derivative of y to the fifth is 5 y to the fourth, derivative of y is 1, leaving us just with 2 x to the fourth, and then from here we can get the double y partial.
06:13
Derivative of y to the fourth would be 4 y cubed, and then the x term is treated as a constant, and so goes to zero, and there we go.
06:25
There are all three second order partial derivatives.
06:30
Moving on, two more.
06:33
Same question, where this time we have the function equal to x times y over x minus y.
06:51
Okay so what will this look like? so first let's do the x partial.
06:57
We need quotient rule, bottom function derivative of the top minus top function derivative of the bottom over bottom function squared.
07:14
Okay and then simplifying the top, you'll notice we'll get x y and then minus x y, that's very fortunate, which leaves us just with negative y squared up top, and then x minus y squared in the denominator.
07:33
So let's go ahead and take the double x partial...