Homework Help: Implicit differentiation of many variables

1. Dec 22, 2016

jonjacson

1. The problem statement, all variables and given/known data

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)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 dzx, then I should do the same calculation for dzy and the total differential of the function z should be:

dz = dzx + dz y

Is this correct?

Last edited: Dec 22, 2016
2. Dec 22, 2016

Staff: Mentor

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

3. Dec 22, 2016

vela

Staff Emeritus
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 $z = \ln (x^2+xy+y^2)$.

4. Dec 23, 2016

jonjacson

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?

No, there is nothing.