Question

(Partial derivative question in which I'm having trouble starting because I don't know how to isolate z):

http://f.imgtmp.com/PRXm0.jpg

Any help would be greatly appreciated!
Thanks in advance!

Answer 1

I would also appreciate the reasoning behind what I must do.

Answer 2

I think because the nature of partials is such that the other variables are treated like constants, \[f_x=4x^3y^7+9x^8z^7+4yz\] so because there is no x being multiplied by the second set of numbers it is solved but is an equation of three variables.

Answer 3

Ok but the answer is not 4+4+9=17 but it is -0.894736842105263

Answer 4

then you take that partial which is equal to zero and solve for z. So then it becomes \[-4x^3y^7=9x^8z^7+4yz\]
\[-4x^3y^7=z^7(9x^8+4y/z^6)\]
\[-[(4x^3y^7)/(9x^8+4y/z^6)]=z^7\]

Answer 5

then you have the seventh root of all that and the negative remains and should come out to what you need I think

Answer 6

I plugged in z^6 as 1^6 and get -0.845 which is wrong, the answer is: -0.894736842105263

Answer 7

That's the best I could think of because isolating the z before differentiating is imposible

Answer 8

Ok, wait I'll try to brainstorm a solution with you. Let me rethink this through.

Answer 9

z = f(x,y) should mean 1 = f(x,y), since we're being asked to find the rate of change on the xy plane, right?

Answer 10

I don't think we can make that conclusion just because we're differentiating within x and y.

Answer 11

because altogether it is the rate of change in F in the direction of x, then y, so our curves should be including variable x values on this surface.

Answer 12

I meant z values

Answer 13

I'm trying to find something in book/notes but I can't but I'm still looking.

Answer 14

(I didn't ignore what you said above.)

Answer 15

I'm trying to see if there is some other way to isolate z before starting myself

Answer 16

Alright. No rush.

Answer 17

I can't see any other way that 4xyz bit really throws things off in terms of isolating z by itself. Without getting it to z I can't see what the surface looks like or anything that could be helpful. It has got me all over it though and I can't leave it alone, but I have no forseeable solution outside of the first thing we tried. If I get something going I'll post it for you.

Answer 18

Alright thanks for your continued effort, I will do some other problems before I go eat and then I'll be back to try this one yet again with a fresh look.

Answer 19

good luck if I don't get something back to you.

Answer 20

Thanks.

Answer 21

you know I was just thinking that the the question stated that near (1,1,1) there is an equation f(x,y). The equation that we start with is in the form w=f(x,y,z). At (1,1,1) there is not a point that exists in our equation because 10 doesn't =7. Maybe that fact has something to do with how we approach the problem. I'm thinking that overall it has to be some sort of error or even a bunk question because because (1,1,1) dne in the initial function we could not possibly have a tangent plane there anyway. Also to reduce the three variable function to term of two, we would have to isolate z by itself which seem impossible to do. I can't call it otherwise, and even my first thought doesn't seem to work in the substitution of values at the point. Sorry if it's misleading or unhelpful but I can't see it working out. If you do solve it please post back, because I would be totally interested. Take care

Answer 22

Hey TuringTest :)

Answer 23

Hey s3a, I'm looking, but I'm pretty tired and it does seem a bit strange.
I think I have a better chance of getting the tangent plane than the rest, but I'll have to get back to you.

Answer 24

Well just the tangent plane is better than nothing unless you're planning to go to bed then I don't want to get in the way.

Answer 25

the answer I'm getting for the tangent plane is part is 0.-8947368421...
I'll be back in a few minutes to explain
basically, you take the gradient and use the components of the gradient of that point to establish an orthogonal vector, from which you can get the tangent plane

Answer 26

You said enough. I got it.

Answer 27

So you don't know for the rest for sure for today? Sorry for all the fors, lol.

Answer 28

sorry I don't right now
maybe tomorrow it'll hit me
hope I helped a bit, g'night :D

Answer 29

You did and I appreciate it as always. Good night.

Answer 30

ok here's an idea:
for the first part take the derivative of everything with respect to x, and use the chain rule on z by remembering that z=f(x,y) in that region\[4x^3y^7+8yz^7\frac{\partial f}{\partial x}+9z^7x^8+7z^6x^9\frac{\partial f}{\partial x}+4yz+4xy\frac{\partial f}{\partial x}=0\]plug in\[(x,y,z)=(1,1,1)\]and solve for partial of f wrt x
repeat the process with the other variables, it should be the same.

I hope that works, good luck!

Answer 31

oh good, and that method gives the answer at the bottom: -17/19
so I have faith in it
I hope you get this answer by the time you need it

Answer 32

Well, unfortunately, I did not get it by the time I needed it but that's my fault really. Having said that though, I still am interested in understanding the entire problem and, mechanically, I fully get how to do it.

What still confuses me is why I can just ignore the fact that there are z letters on both sides of the equation. My first thought was: "Just treat z as a constant always." but if I did that then why is z = f(x,y) where f(x,y) is a function that itself varies. If the z letters inside f(x,y) are constant, shouldn't the z that is equal to f(x,y) also be constant? I think I'm currently looking for a deeper theoretical answer if you can provide one.

Answer 33

we are told that z is a function x and y in that region, hence we have to use the chain rule\[\frac{\partial}{\partial x}z=\frac{\partial}{\partial x}f(x,y)=\frac{\partial f}{\partial x}\]because we are taking the derivative with respect to a variable that z is dependent on

Answer 34

it's like implicit differentiation in a way...

Answer 35

Oh my god, wait!

Answer 36

rather\[\frac{\partial}{\partial x}g(z)=g
_x(z)\frac{\partial z}{\partial x}\]sorry just felt like posting that

Answer 37

I just realized that big equation with x, y and z letters is = 7 and not = 7. And z is a function of other variables so it's not "like" implicit differentiation, it IS implicit differentiation.

Tell me if my epiphany is wrong lol.

Answer 38

and not = z I meant

Answer 39

Oh, I didn't get what you meant by z on both sides
yeah it's a seven, and yeah I guess it is implicit differentiation
I just never have seen this type of question before, so I don't want to misuse terminology

Answer 40

I suppose these questions are designed to test the fundamental understanding of the calculus involved. Anyways, I think I fully get it now! If I realize that this is not the case, I will tell you but I think it is. Thanks again :D.

Answer 41

very welcome :D