导数的定义函数y=f(x),如果当时,有极限,就说函数y=f(x)在点x0处可导,并把这个极限叫做f(x)在点x0处的导数(或变化率),记做0xxy0|)('0'xxyxf或xxfxxfxyfxfxxxx)()(limlim|)(0000'0'0000)()(lim)(0xxxfxfxfxx下式表示:事实上,导数也可以用.)0(||1的导数数:利用导数的定义求函例xxy;1lim,1)(,,0|,|0xyxxxxxyxyxxyx则时当解:;1lim,1)()(,,00xyxxxxxyxyxx时当.0101xxy练习1:函数f(x)=|x|(1+x)在点x0=0处是否有导数?若有,求出来,若没有,说明理由.0)(0)()()0()0(,)0()0()(:2222xxxxxxxffxfyxxxxxxxf故有显然解,1)1(limlim00xxyxx.1)1(limlim00xxyxx.,0,limlim00无极限时xyxxyxyxx故函数f(x)=|x|(1+x)在点x0=0处没有导数,即不可导.练习2:证明:(1)可导的偶函数的导函数为奇函数;(2)可导的奇函数的导函数为偶函数.证:(1)设偶函数f(x),则有f(-x)=f(x).).()()(lim,)(0xfxxfxxfxfyx可导函数).()()(lim)()(lim)()(lim)(000xfxxfxxfxxfxxfxxfxxfxfxxx.)(立是奇函数,从而命题成xf(2)仿(1)可证命题成立,在此略去,供同学们在课后练习用.5.导数的几何意义:曲线y=f(x)在点P(x0,,f(x0))的切线的斜率.即曲线y=f(x)在点P处的切线的斜率是f’(x0)6.利用导数求曲线的切线方程7.导数与切线的关系(1)f’(x0)>0切线的斜率大于0.(2)f’(x0)<0切线的斜率小于0.(4)f’(x0)不存在,切线的斜率不存在.(3)f’(x0)=0,切线的斜率等于0.例2:如图,已知曲线,求:(1)点P处的切线的斜率;(2)点P处的切线方程.)38,2(313Pxy上一点yx-2-112-2-11234OP313yx.])(33[lim31)()(33lim3131)(31limlim,31)1(2220322033003xxxxxxxxxxxxxxxxyyxyxxxx解:.42|22xy即点P处的切线的斜率等于4.(2)在点P处的切线方程是y-8/3=4(x-2),即12x-3y-16=0.例2:已知曲线方程为y=x2,求过B(3,5)点且与曲线相切的直线方程2x-y-1=0;10x-y-25=0