Can you please explain how to differentiate $f(x,y)$ with respect to $x-y$ ?
Thank you very much.
Can you please explain how to differentiate $f(x,y)$ with respect to $x-y$ ?
Thank you very much.
For a general co-ordinate $u$, $\frac{df}{du} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial u} + \frac{\partial f}{\partial y}\frac{\partial y}{\partial u}$
So for $u = x-y$, we have:
$\frac{df}{du} = \frac{\partial f}{\partial x} - \frac{\partial f}{\partial y}$
Note that the derivative given above is different to the partial derivative $\frac{\partial f}{\partial u}$ which will in general depend on a given parameterisation of the curve as $f = f(u,v)$, as mentioned in another answer.
Depends on what the other variable in your coordinate system is. For example, if your new system is $u = x - y$, $v = x$ and $f(x, y) = x + y$, then $ f(u, v) = -u + 2v $ and so the derivative with respect to $u$ is $-1$. But if the new system is $u = x - y$, $v = y$, then $ f(u, v) = u + 2v $ and the derivative is $1$.