高数第二章导数和微分17-24课讲义-赵老师【公众号：小盆学长】免费分享.pdfVIP免费

赵军高等数学零基础课程一、导数的定义定义1.设函数)(xfy在点0x0limxx00)()(xxxfxfxyx0lim)()(0xfxfy0xxx存在,)(xf并称此极限为)(xfy记作:;0xxy;)(0xf;dd0xxxy0d)(dxxxxf即0xxy)(0xfxyx0limxxfxxfx)()(lim000hxfhxfh)()(lim000则称函数若的某邻域内有定义,在点0x处可导,在点0x的导数.0limxx00)()(xxxfxfxyx0lim)()(0xfxfy0xxx不存在,就说函数在点不可导.0x若0lim,ΔΔΔxyx也称)(xf在0x若函数在开区间I内每点都可导,此时导数值构成的新函数称为导函数.记作:;y;)(xf;ddxy.d)(dxxf注意:)(0xf0)(xxxfxxfd)(d0就称函数在I内可导.的导数为无穷大.若极限例1.求函数Cxf)((C为常数)的导数.解:yxCCx0lim0即0)(C例2.求函数)()(Nnxxfn.处的导数在ax解:axafxf)()(axlim)(afaxaxnnaxlim(limax1nx2nxa32nxa)1na1nanxxfxxf)()(0limx说明：对一般幂函数xy(为常数)1)(xx例如，)(x)(21x2121xx21x1)(1x11x21x)1(xx)(43x4743xhxhxhsin)sin(lim0例3.求函数xxfsin)(的导数.解:,xh令则)(xfhxfhxf)()(0limh0limh)2cos(2hx2sinh)2cos(lim0hxh22sinhhxcos即xxcos)(sin类似可证得xxsin)(cosh则令,0hxt原式htfhtfh2)()2(lim0)(lim0tfh)(0xf是否可按下述方法作:例4.设)(0xf存在,求极限.2)()(lim000hhxfhxfh解:原式0limhhhxf2)(0)(0xfhhxf2)(0)(0xf)(210xf)(210xf)(0xf)(2)(0hhxf)(0xf例5.证明函数xxf)(在x=0不可导.证:hfhf)0()0(hh0h,10h,1hfhfh)0()0(lim0不存在,.0不可导在即xx在点0x的某个右邻域内单侧导数)(xfy若极限xxfxxfxyxx)()(limlim0000则称此极限值为)(xf在处的右导数,0x记作)(0xf即)(0xfxxfxxfx)()(lim000(左)(左))0(x)0(x))((0xf0x例如,xxf)(在x=0处有,1)0(f1)0(f定义2.设函数有定义,存在,xyOxy定理.函数在点0x)(xfy,)()(00存在与xfxf且)(0xf.)(0xf)(0xf存在)(0xf)...