Implicit differentiation of many variables

[jonjacson]

Homework Statement

For the given function z to demonstrate the equality:

[/B]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)² - z² = ( (x+dx) * (z+dz) + y * (z+dz)³ ) - (x * z + y * z³)

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?

 

[Mark44]

Homework Statement

For the given function z to demonstrate the equality:

[/B]

Is there information missing from the image above, especially in the upper right corner?

 

[vela]

Homework Statement

[/b]
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)

This result doesn't match what's given in the solution, so why do you think your guess for $z$ is correct?

The solution is definitely wrong for the given $z$, but your guess is wrong too. It looks like the solution used $z = \ln (x^2+xy+y^2)$.

 

[jonjacson]

This result doesn't match what's given in the solution, so why do you think your guess for $z$ is correct?

The solution is definitely wrong for the given $z$, but your guess is wrong too. It looks like the solution used $z = \ln (x^2+xy+y^2)$.

Thanks for your answer.

But with my guess I find:

∂z/∂x= 2x/(x^2 + y^2)

∂z/∂y= 2y/(x^2 + y^2)

So if I substitute in the equation I get:

x * (2x/(x^2 + y^2)) + y * (2y/(x^2 + y^2)) = 2

In the denominator we have the same functions, so we can simply sum the numerators to get:

(2 x^2 + 2 y^2 )/ (x^2 + y^2) = 2

And this equation is true. What am I doing wrong?

Is there information missing from the image above, especially in the upper right corner?

No, there is nothing.