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HMMT 二月 2002 · CALC 赛 · 第 3 题

HMMT February 2002 — CALC Round — Problem 3

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. We are given the values of the differentiable real functions f, g, h , as well as the derivativesof their pairwise products, at x = 0:
    ′ ′ ′
    f (0) = 1; g (0) = 2; h (0) = 3; ( gh ) (0) = 4; ( hf ) (0) = 5; ( f g ) (0) = 6 .
    ′
    Find the value of ( f gh ) (0).
解析

英文解析

  1. We are given the values of the differentiable real functions f, g, h , as well as thederivatives of their pairwise products, at x = 0:
    ′ ′ ′
    f (0) = 1; g (0) = 2; h (0) = 3; ( gh ) (0) = 4; ( hf ) (0) = 5; ( f g ) (0) = 6 .
    ′
    Find the value of ( f gh ) (0).
    ′ ′ ′ ′ ′ ′
    Solution: 16 By the product rule, ( f gh ) = f gh + f g h + f gh = (( f g ) h + ( gh ) f +
    ′
    ( hf ) g ) / 2 . Evaluated at 0, this gives 16.