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Is this true?

Mixed partial derivatives are always equal

No — it is false.

The counterexample

This one comes with the Notes pack.

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85 counterexamples are free — including most of the ones Part C leans on hardest.

The kind of mistake this is

Converse assumed

The theorem runs one way. You used it in the other.

Drill statements like this
multivariable

Others that fail the same way

From Functions of Several Variables › Partial derivatives, differentiability, chain rule

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