Partial derivative holding something constant

Question

I was reading this question Manipulating Partial Derivatives of Inverse Function when a doubt raise up. Someone can explain how the "user147263" developed the partial derivative holding $eta-xi$ constant? That is:
$$xi = x - y qquad eta = x+y$$
$$x = frac12(xi+eta) qquad y = frac12(eta-xi)$$
$$dfrac {partial x} {partial xi}bigg|_{eta-xi text{ constant}} = 1$$
How to develop the third partial derivative?

Äres · Accepted Answer

As you want to hold $y$ constant and $y=frac{1}{2}(eta-xi)$, then $(eta-xi)$ must be held constant.
Since $$x = frac12(xi+eta)=frac{1}{2}(eta-xi+2xi)=xi+frac{1}{2}(eta-xi)$$
and $(eta-xi)$  is constant then it follows that $$dfrac {partial x} {partial xi}bigg|_{eta-xi text{ constant}} = 1$$

Partial derivative holding something constant

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