# Interchange of evaluation in double partial differentiation

Is it true that

$\frac{\partial}{\partial y} ( [\frac{\partial}{\partial x} F(x,y)]|_{x=0}) = [\frac{\partial}{\partial y} ( \frac{\partial}{\partial x} F(x,y))]|_{x=0}$

?

Seems easy but I can't affirm whether it is true or not.

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It is. When differentiating with respect to $y$, we hold $x$ constant anyway. So we could first set $x=\text{constant}$ and then set the constant to zero, or just set $x$ to zero at once. – user31373 Sep 23 '12 at 2:30
@LVK: You could turn that comment into an answer. – joriki Sep 23 '12 at 2:31

It is. When differentiating with respect to $y$, we hold $x$ constant anyway. So we could first set $x=\text{constant}$ and then set the constant to zero, or just set $x$ to zero at once.