文档介绍：导数一、导数定义式的几种等价形式0limxx?00)()(xxxfxf??xxfxxfx???????)()(lim000hxfhxfh)()(lim000????0limxyx? ????0( )f x??左导数、右导数:0000( ) ( )( ) limx xf x f xf xx x???????0000( ) ( )( ) limx xf x f xf xx x???????二、( ) ( )limx xf x f xx x???0 00( ) ( )limhf x h f xh?? ?或是否存在考虑左右导数0 0( ), ( )f x f x? ?? ?是否都存在且相等考虑是否不连续(连续不一定可导,但不连续一定不可导!)三、必须用定义求导数的情形1. . . 仅知函数在一点可导,【注】某些“乘积型”的复杂函数用定义求导较方便。( )f x不知在该点的附近(一个邻域)、常数和基本初等函数的求导公式??)(C0??)(?x1???x??)(sinxxcos??)(cosxxsin???)(tanxx2sec??)(cotxx2csc???)(secxxxtansec??)(cscxxxcotcsc?特别地:)(?xx21????x121x????)(xaaaxln??)(xexe??)(logxaaxln1??)(lnxx1??)(arcsinx211x???)(osx211x????)(arctanx211x???)cot(arcx211x??(ln | |)x??,+ 、-、×、÷、求导数的主要法则六、导数的几何意义)(xfy?在点),(00yx的切线斜率0( )f x?切线方程:))((000xxxfyy????法线方程:)()(1000xxxfyy?????注:xyo( )y f x?C),(、求高阶导数的主要方法(1)逐次求导归纳法;(2)n 阶导数的公式及求导法则;注:求一点处高阶导数的好方法------函数的幂级数展开(以后学)( )0( )nf ( )( )ax n n axe a e?(1)( )(sin ) sin(n nax a ax? ?( )(cos ) cos(n nax a ax? ?)2??n)2??n(2)(3)( )11 ( 1) !( )( )nnnnx a x a?? ??? ?(4)( )( 1) ( 1) ,( ) !0k nk nk k k n x n kx n n kn k??? ????? ??????(k为正整数。)(a 为常数)