A mixed partial derivative is a partial derivative of second or greater order taken with respect to at least two different variables. With subscripts recording the order of differentiation from left to right,
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Thus 
differentiates first with respect to 
and then with respect to 
, whereas the operators in Leibniz notation act from
right to left (Lebl 2026, p. 60).
If 
is twice continuously differentiable
in an open set, then 
throughout that set. More
generally, if all partial derivatives of order
at most 
are continuous, changing the order of 
differentiations does not change the result (Lebl 2026, pp. 60-62). In particular,
the Hessian of a twice continuously
differentiable real-valued function is a symmetric
matrix.
Existence of both mixed partial derivatives at a point does not ensure their equality. They can differ even for a continuously differentiable function, as in the partial derivative counterexample discussed by Trott (2006).
