HMMT 二月 2002 · CALC 赛 · 第 3 题
HMMT February 2002 — CALC Round — Problem 3
题目详情
- We are given the values of the differentiable real functions f, g, h , as well as the derivatives 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).
解析
- We are given the values of the differentiable real functions f, g, h , as well as the derivatives 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.