y=1⼀(x^2-2x+4)二阶导数怎么求

y=1/(x^2-2x+4)二阶导数计算技巧

y = 1/(x² - 2x + 4)

dy/dx = -(x² - 2x + 4)'/(x² - 2x + 4)²

= -(2x - 2)/(x² - 2x + 4)² = 2(1 - x)/(x² - 2x + 4)²

d²y/dx² = 2 • [(x² - 2x + 4)²(1 - x)' - (1 - x)(x² - 2x + 4)²']/(x² - 2x + 4)⁴

= 2 • [(x² - 2x + 4)²(-1) - (1 - x) • 2(x² - 2x + 4) • (2x - 2)]/(x² - 2x + 4)⁴

= 2 • (- x² + 2x - 4 + 4x² - 8x + 4)/(x² - 2x + 4)³

= 2 • (3x² - 6x)/(x² - 2x + 4)³

= 6x(x - 2)/(x² - 2x + 4)³

概念分析

二阶导数是比较理论的、比较抽象的一个量，它不像一阶导数那样有明显的几何意义，因为它表示的是一阶导数的变化率。在图形上，它主要表现函数的凹凸性，直观的说，函数是向上突起的，还是向下突起的。

y = 1/(x² - 2x + 4)
dy/dx = -(x² - 2x + 4)'/(x² - 2x + 4)²
= -(2x - 2)/(x² - 2x + 4)² = 2(1 - x)/(x² - 2x + 4)²
d²y/dx² = 2 • [(x² - 2x + 4)²(1 - x)' - (1 - x)(x² - 2x + 4)²']/(x² - 2x + 4)⁴
= 2 • [(x² - 2x + 4)²(-1) - (1 - x) • 2(x² - 2x + 4) • (2x - 2)]/(x² - 2x + 4)⁴
= 2 • (- x² + 2x - 4 + 4x² - 8x + 4)/(x² - 2x + 4)³
= 2 • (3x² - 6x)/(x² - 2x + 4)³
= 6x(x - 2)/(x² - 2x + 4)³