# Differentiating with respect to a function using variable transformation

Earlier I asked a question about differentiating $f(x,y)$ with respect to $x-y$. I am working on the solutions trying to use the hints from earlier questions.

Is it correct to do the following: Define $u = x-y$ and $v = x + y$. Rewrite $f(x,y)$ as $g(u,v)$. Now, I understand that taking partial of $g(u,v)$ wrt $u$ is not enough because it keeps $v$ fixed, but my goal is to find how does $f$ change when $x-y$ changes without other restrictions. Can I then take total differential of $g(u,v)$ to answer that?

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What does "differentiating $f(x,y)$ with respect to $x-y$" mean? – wj32 Dec 21 '12 at 4:06
Why $x+y$? This is the issue. You could choose many other complementary variables. A partial derivative should hold some variable besides $x-y$ fixed. Generally it will matter what the second variable is. – James S. Cook Dec 21 '12 at 4:41
My goal is to answer a question "how does f(x,y) change when x-y changes". What is a proper mathematical way to address it? – Essa Dec 21 '12 at 12:50
In particular, I am looking at f(x,y) = x^2 + y^2 and I want to understand the sensitivity of it to g(x,y) = x-y. – Essa Dec 21 '12 at 13:58