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此问题历属教学中的难点之一。构成疑难的原因很多,其中以自变量个数增多、中间变量个数不等、复合层次有异为主。本文就此问题谈点体会。 定理 若函数υ_i=i(x_1,x_2,…,x_m)i=1,2,…,n,在(x_1,x_2,…,x_m)点有偏导数,函数Z=f(u_1,u_2,…,u_n)在对应点(u_1,u_2,…,u_n)可微,则复合函数 Z=f〔1,(x_1,x_2,…,x_m),……,n(x_1,x_2,…,x_n)〕在(x_1,x_2,…,x_n)点有偏导数,并且
This problem is one of the difficulties in teaching. There are many reasons for the difficulty. Among them, the number of independent variables increases, the number of intermediate variables varies, and the composite levels vary greatly. This article talk about this issue experience. Theorem If the function υ_i = i (x_1, x_2, ..., x_m) i = 1,2, ..., n, there is a partial derivative at (x_1, x_2, ..., x_m), the function Z = f (u_1, u_2, ... , u_n) are differentiable at corresponding points (u_1, u_2, ..., u_n), the complex function Z = f [1, (x_1, x_2, ..., x_m), ..., n (x_1, x_2, ..., x_n) ] Has partial derivatives at (x_1, x_2, ..., x_n), and