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第二节 导数运算与导数公式 Rules for Finding Derivatives and the Derivative Formulas

第二节 导数运算与导数公式 Rules for Finding Derivatives and the Derivative Formulas. 一、函数的和、差、积、商的求导法则 Rule for Finding Derivatives of the Sum, Difference, Product and Quotient of Two Functions. 二、反函数的求导法则 Rule for Derivative of Inverse Functions. 三、导数基本公式 the Derivative Formulas. 返回.

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第二节 导数运算与导数公式 Rules for Finding Derivatives and the Derivative Formulas

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  1. 第二节 导数运算与导数公式 Rules for Finding Derivatives and the Derivative Formulas 一、函数的和、差、积、商的求导法则 Rule for Finding Derivatives of the Sum, Difference, Product and Quotient of Two Functions 二、反函数的求导法则 Rule for Derivative of Inverse Functions 三、导数基本公式 the Derivative Formulas 返回

  2. 一、函数的和、差、积、商的求导法则Rules for Finding Derivatives and the Derivative Formulas 定理

  3. (1) (2) ( ) (3) Theorem 1: If the function f=f(x) ang g=g(x) are differentiable at x,then the sum,difference, product and quotient (v(x)≠0) of the two functions are differentiable at x ,and

  4. 于是法则(2)获得证明. 法则(2)可简单地表示为 证(2)

  5. 证(3)

  6. (C 为常数)时,有 在法则(2)中,当 于是法则(3)获得证明. 法则(3)可简单地表示为

  7. 例1（1） 解 同理可得 返回

  8. （2） 解 同理可得

  9. 解 （4） 求 及 解 （3）

  10. （5） 求 解

  11. 二、反函数的求导法则Rule for Derivative of Inverse Functions 定理 或 即 反函数的导数等于直接函数导数的倒数.

  12. Theorem 2: If the function x=f(y) is monotonic defferentiable on the interval Ⅰ and f′( y)≠0,then its inverse function is differentiable on the interval I and .

  13. 证 于是有

  14. 例2(1) 同理可得 解

  15. (2)

  16. (3) 解 特别地 返回

  17. 三、求导法则与导数基本公式Rules for Finding Derivatives and the Derivative Formulas 1.常数和基本初等函数的导数公式 the Derivative Formulas of the Constant Function and the Basic Elementary Functions

  18. = = u u ( x ), v v ( x ) 设 都可导，则 ¢ ¢ ¢ ¢ ¢ = = ( 是常数) 1 , 2 ( u v ) u v ( cu ) c u （ ） （ ） ¢ ¢ - u u v u v ¢ ¢ ¢ = + 3 , 4 . （ ） （ ） ¢ ( uv ) u v u v = ¹ ( ) ( v 0 ) 2 v v 2.函数的和、差、积、商的求导法则

  19. 或 3.反函数的求导法则

  20. 例3设 , 计算 . 例4设 ,计算 . 解： = . 若 , 怎么办？

  21. 例5 ，求f `(x) 解： 例6 ，求f `(x).

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