Thank you for the explanations. I agree with you that this more abstract view of the chain rule (and the derivative itself) is superior to the sum-of-products f

Thank you for the explanations. I agree with you that this more abstract view of the chain rule (and the derivative itself) is superior to the sum-of-products formula one usually sees in a first course in multivariable calculus, but I feel most students have to learn the complicated, technical version first before they can see the beauty of the more abstract one.