由定义求导数步骤:(1)求增量Ay = f(x+ △r)- f()Ay _ f(x+△r)- f(x)(2)算比值AxAxAy(3)求极限y'= limAr-0 △x例1 求函数 f(x)=C(C为常数)的导数,f(x+h)-f(x)C-C解 f'(x)=lim= 0.lim-hh-→0hh-→0即(C)' = 0
由定义求导数 步骤: 例1 求函数 f (x) = C(C为常数)的导数. 解 h f x h f x f x h ( ) ( ) ( ) lim 0 + − = → h C C h − = →0 lim = 0. 即 (C) = 0
例2设函数 f(x)= sin x,求(sin x)及(sin x)元4sin(x + h)- sin x解(sinx)=limhh→0hsin2h= lim cos(x +=cosx.h2h-→02即(sin x)' = cos x.V2.. (sin x)=cosx元2A
例 2 ( ) sin , (sin ) (sin ) . 4 = = x 设函数 f x x 求 x 及 x 解 h x h x x h sin( ) sin (sin ) lim0 + − = → 2 2 sin ) 2 limcos( 0 hh h x h = + → = cos x. 4 4 2 (sin ) cos . x x 2 x x = = = =
例3 求函数 y=x"(n为正整数)的导数(x+h)" -xn解(x")'=limhh-→0n(n-1)= lim[nx"-1x"-2 h + ... + h"-I | = nx"-1+2!h→0即(x")'= nx"-1一般的(x")'= uxu-1(uER)(Vx)==x2例如,22Vx1(x-') =(-1)x-1-1
例3 求函数 y x (n为正整数)的导数. n = 解 h x h x x n n h n + − = → ( ) ( ) lim 0 ] 2! ( 1) lim[ 1 2 1 0 − − − → + − = + n n n h x h h n n nx −1 = n nx ( ) . −1 = n n 即 x nx 1 1 2 1 1 ( ) . 2 2 x x x − 例如, = = 1 1 1 2 1 ( ) ( 1) . x x x − − − = − = −
例4 求函数 f(x)=a(a>0,a±1)的导数at+hat解(a")=limhh-→01-1aXlim=ahh→0=axIn a.即(e*)'=e*(a)= aIna
例 4 求函数 f (x) = a (a 0,a 1)的导数. x 解 h a a a x h x h x − = + →0 ( ) lim h a a h h x 1 lim0 − = → a ln a. x =