How to Approach Proving a Partial Derivative Homework Problem?

[xortan]

Homework Statement

I have attached a picture of the problem. The question is the first one.

Homework Equations

The Attempt at a Solution

I tried subbing u and v into the right hand side of the equation. I expanded and simplified but I do not think that is the right way to go about it. If somebody can point me in the right direction that would be very helpful. Thank you!

 

[tiny-tim]

hi xortan!

I tried subbing u and v into the right hand side of the equation.

my guess is that it's quicker to start from the left hand side

(you could also "factor" it into (∂/∂x + ∂/∂y)(∂/∂x - ∂/∂y)z)

… show us what you get ​

 

[xortan]

Well I am really new at partial derivative, we just started this course. Was doing a single variable calc course last semester and only touched on this stuff briefly at the end. I am not understanding where to go when starting from the left hand side. I asked another one of my instructors and he showed me this "brute force" way of doing it. I got it to a point of



(∂²z/∂x² = (∂/∂x)(∂z/∂u)2x + ∂z/∂u * 2 + (∂/∂x)(∂z/∂v)2y

I got to deal with the first term now but the question seems to be blowing up and I know there has to be an easier way of doing this