Interview Question: Chaining in Calculus

Sample Question #313 (math)
 
(A real-world interview question)
 
Formally state the Chain Rule. Can you sketch a proof?
 
 
[Tip: when someone asks you to "formally" state a theorem, he/she expects you to mention the conditions under which the main assertion holds, e.g., in the case of the chain rule — you do remember the chain rule from Calculus 101, don’t you? — you need to mention the regularity conditions of differentiability.]
 
[Hint: the proof hinges on the definition of a derivative and taking limits. When you sketch a proof, you do not need to be rigorous, just need to tell the interviewer what the major steps are.]
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