For example, given the equation $x + y - z + \cos(xyz) = 0$. Is it possible to find partials of $z$ w.r.t. $x$ and $y$ at the point $(0,0,0)$?
2026-03-25 17:43:34.1774460614
Can implicit derivatives exist at points where an equation is not satisfied?
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No. You need an equation to relate $z$ to $x$ and $y$, and so the point in question must satisfy the equation. The only possible way to make sense of your question is to consider all the different level surfaces $x+y-z+\cos(xyz)=c$ (as $c$ varies). The origin will lie on such a surface for $c=1$. Now you must check that the hypotheses of the implicit function theorem hold at the point on that surface before proceeding with your question. (They do.)