This approach clarifies the chain rule's application, suggesting the need for better teaching methods.
One common proof of the chain rule constructs an auxiliary function and causes some difficulty for learning and teaching. Another common proof based on the notion of the differential fails in its existing form. A new proof is given, which directly faces the subtle point that dependent variable and independent variable have different behaviours in the definition of the limit of a function.
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Zhibin Yan (2025) studied this question.
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