- #1

jonjacson

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## Homework Statement

For the given function z to demonstrate the equality:

**As you see I show the solution provided by the book, but I have some questions on this.**

I don't understand how the partial derivative of z respect to x or y has been calculated.

Do you think this is correct?

I think this is a giant errata, I guess the function z is not given implicitly and it simply is:

z = ln ( x ^2 + y^2)

The partial derivatives are calculated normally:

∂z/∂x= 2 * x/(x^2 + y^2)

Similar for y, and with this it is straighforward to demonstrate the equality.

What do you think? There are two options:

1.- Or the statement and solution of the given problem is correct---> In that case I don't understand anything. Could you explain how to get the partial derivatives?

2.- Or there is a giant errata, z is not given implicitly and the calculation is easy.

And forgeting this problem I was wondering in case I found an equation with z given implicitly like:

z^2 = x * z + y * z^3

How would we differenciate this equation?

As we have many variables we should choose which are maintained constant and which are changing. Suppose we differenciate this expression considering x is changing, y is constant but z obviously changes, due to the changes in x.

The receipt is changing x for x+dx, z changes to z+dz and y doesn't change at all. I get:

(z+dz)

I don't understand how the partial derivative of z respect to x or y has been calculated.

Do you think this is correct?

I think this is a giant errata, I guess the function z is not given implicitly and it simply is:

z = ln ( x ^2 + y^2)

The partial derivatives are calculated normally:

∂z/∂x= 2 * x/(x^2 + y^2)

Similar for y, and with this it is straighforward to demonstrate the equality.

What do you think? There are two options:

1.- Or the statement and solution of the given problem is correct---> In that case I don't understand anything. Could you explain how to get the partial derivatives?

2.- Or there is a giant errata, z is not given implicitly and the calculation is easy.

And forgeting this problem I was wondering in case I found an equation with z given implicitly like:

z^2 = x * z + y * z^3

How would we differenciate this equation?

As we have many variables we should choose which are maintained constant and which are changing. Suppose we differenciate this expression considering x is changing, y is constant but z obviously changes, due to the changes in x.

The receipt is changing x for x+dx, z changes to z+dz and y doesn't change at all. I get:

(z+dz)

^{2}- z^{2}= ( (x+dx) * (z+dz) + y * (z+dz)^{3}) - (**x * z + y * z**

After neglecting diferentials of order two and three I get:

dz = dx * (z dx / 2x - x -3 y z^2)

But this differential arose because there was a change on x, so I should call it dz

dz = dz

Is this correct?

^{3})After neglecting diferentials of order two and three I get:

dz = dx * (z dx / 2x - x -3 y z^2)

But this differential arose because there was a change on x, so I should call it dz

_{x}, then I should do the same calculation for dz_{y}and the total differential of the function z should be:dz = dz

_{x}+ dz_{y}Is this correct?

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