HELP: Partial Derivative exercise (Implicit)

[DaveMat]

Hello, I'm new to the forum, I think this site is great, and I hope that you could help me with this exercise, I don't know how to write the problem with languages like Latex, I just used a Latex Editor online, so please excuse me for any errors or if I don't understand, and if there is an easier way to write the problems in a "mathematical way" please tell me how. Now the exercise:

If

defines z like an implicit function of x,y. Prove that:

I managed to do these:

I can do the first derivative just fine, but I still have problems with higher order partial derivatives.


P.S.
I used this: https://www.codecogs.com/latex/eqneditor.php?lang=en-us
to generate the pictures

 

[Ishuda]

Hello, I'm new to the forum, I think this site is great, and I hope that you could help me with this exercise, I don't know how to write the problem with languages like Latex, I just used a Latex Editor online, so please excuse me for any errors or if I don't understand, and if there is an easier way to write the problems in a "mathematical way" please tell me how. Now the exercise:

If

defines z like an implicit function of x,y. Prove that:

I managed to do these:

I can do the first derivative just fine, but I still have problems with higher order partial derivatives.


P.S.
I used this: https://www.codecogs.com/latex/eqneditor.php?lang=en-us
to generate the pictures

Suppose
g(x,y) = $\displaystyle \dfrac{1}{3z^2(x,y)\, -\, x}$

What is $\displaystyle \dfrac{\partial\, g}{\partial\, x}$?