对复合函数求导定理证明的评论与研究

本文指出了一些广泛使用的教科书对复合函数求导定理的证明是不严格,并创造性地举例阐明存在这样的复合函数f(u),它对自变量u可导,而对中间变量u=g(x)不可导。本文也给出了该定理的一个严格证明,同时揭示了复合函数求导法则的表达形式与其内容并不完全一致。

Comments and Research on Proofs of Chain Rule

In this paper we point out that the proofs of Chain Rule in many intensively used textbooks are not strict an construct anexa mple of a composite function f(u) which is differentiable with respects t o the independednt variable u, but is not differentiable with respect to the dependent variable u=g(x). A strict proof of Chain Rule is presented. Incon sistency of the form and content of Chain Rule is disclosed.