第二节 求导法则.pptVIP免费

1第二节求导法则xxuxxux)()(lim0xxvxxvx)()(lim0)()(xvxu.设)(xuu,)(xvv可导,则vu,uv,vu)0(v均可导,且有)(vuxxvxuxxvxxux)]()([)]()([lim0证.)()1(vuvu注:可推广到有限多个函数的和与差。一、导数的四则运算法则2因为)(xv可导,必连续,xvxuxxvxuxyxxxx0000lim)()(limlimlim)()()()(xvxuxvxu.故)()(lim0xvxxvx,于是)()()()(xvxuxxvxxuy)]()()()([xxvxuxxvxxu)]()()()([xvxuxxvxuvxuxxvu)()(,证.)()2(vuvuuv3证略.特别,.)()3(2vvuvuvu.)1(2vvv1.uCCu)(；2.可推广到有限多个函数的乘积，如推论wuvwvuvwuuvw)(.)()2(vuvuuv证wuvwuvuvw)()()(wuvwvuvu)(.wuvwvuvwu4求下列函数的导数：例1xxxxysin4523.123;cos45492xxxyxxy3.22;3ln3322xxxxyxxyxcose.32xxyxcose24.)50()2)(1()(xxxxf,求)2(f.2)50()4)(3)(1()2(xxxxxf!48xxxcose2;sine2xxx5例2.tan的导数求xy解)(tanxyxxxxx2cos)(cossincos)(sinxxx222cossincosx2cos1xx2sec)(tanxx2csc)(cot类似可得即)cossin(xx，x2sec6例3.sec的导数求xy解)(secxyxx2cos)(cos.tansecxxxx2cossinxxxcotcsc)(csc类似可得即xxxtansec)(sec)cos1(x7例4.cos1sin5的导数求xxy解例5.sectan的导数求xxxy解2)cos1(sinsin5)cos1(cos5xxxxxy2)cos1()1(cos5xx.cos15x.tansecsectan212xxxxxxy8二、反函数的求导法则定理即反函数的导数等于直接函数的导数的倒数..)(1)(,)(,0)()(yxfIxfyyIyxxy且有内也可导在对应区间那末它的反函数且内单调、可导在某区间如果函数9例6.log的导数求函数xya解,logxya设axxaln1)(log,yax即所以yxxydd1ddaayln1.ln1ax特别，xx1)(ln10例7.arcsin的导数求函数xy解,)2,2(sin内单调、可导在yIyx,0cos)(sinyy且内有在)1,1(xI)(sin1)(arcsinyxycos1y2sin11.112x；211)(arccosxx类似可得211)(arctanxx211...