高數(shù)復(fù)習(xí)-導(dǎo)數(shù)與微分

導(dǎo)數(shù)與微1f23，則limf23hf2h4f2 2f12，則limf1f1hf1 3ycosexy0sin14ysinx3，則dy3x2cosx3dx5fxx3lnxf156y6e2xln3y87yn2sin7xyn49sin7x8ye2xcos2xx10y10210e2x210cos2x1010!210e2x210cos2x 2 y12212e2x212cos2x12212e2x212cos 2 9、設(shè)yefx2fex2，則dy2xefx2fx22xfex210yex2x0y3y311、fxxx1x2x3......x10，則f1fxx1xx2x x f111213110012可微可導(dǎo)是連續(xù)的充分 連續(xù)是極限存在的充分 極限存在是連續(xù)的必 條件 連續(xù)是可微可導(dǎo)的必 條13yxx1x2x2x14、zlnxy，

；f1,0 2x

15fxyx24xy3y2，則limf1yhf1yf1y4616zln2x，則y

dx1217z

23xy

2，則

2 6

2；yx18、uxyz，則duyzxyz1dxxzxyz1dyxyzlnxy19zy在點(diǎn)2,1處當(dāng)x0.1y0.2時(shí)的zf2.1,1.2f2,1xdz

ydx1dydz10.110.2 20zfxx2y2f1f2xf

；fxf2yf y

y2 21（1）fx,y在點(diǎn)x,y處可微分是在該點(diǎn)連續(xù)的充 條件fx,y在點(diǎn)x,y處連續(xù)是在該點(diǎn)可微分的必 條fx,y在點(diǎn)x,y處兩偏導(dǎo)數(shù)存在是在該點(diǎn)處可微分 必 條件fx,y在點(diǎn)x,y處可微分是在該點(diǎn)處兩偏導(dǎo)數(shù)存在 充 條fx,y在點(diǎn)x,y處兩偏導(dǎo)數(shù)存在且連續(xù)是在該點(diǎn)處可微分的 2 2fx,y在點(diǎn)x,y處兩二階混合偏導(dǎo)xy,yx連續(xù)是該兩混合偏導(dǎo)相等的 x2cos

3 3

z22、曲線y2sint上對應(yīng)于t 處的切線方

x3y1 z

法平面方程x

3

3y13z 2 23、曲面ez2zxy7在點(diǎn)2,303x22y31z003x2yz120 x2y3z0 1yeaxexa

a0a1，【解yeaxaxexaxaeaxea1eaxaxlnaexaaxa1eaxea2ye3xcos2xln3x2sin3，求3x 33x23ln3x【解：y3e3xcos2x2e3xsin2x3x 03x3e3xcos2x2e3xsin2x33ln3x3x13、yxsinx x23x，求1x2yx2

esinxlnxx23x2sinxlnx

sinx cosxlnx

2x3x2x24ysin2xx2ysin2xx2ex2ex2lnsin2x2xlnsin2x2x2cotx2 x3x2 x311 2x1）lnylnx21ln2）等式兩邊同時(shí)對x x13 x 1

3 2 1 22x 32 1 6yyxexey1cosxy：1）x0y2）exey1cosyexysinxyy0eyxsin7、求由方程cosyxsinxy所確定的函數(shù)yyx的導(dǎo)數(shù)

x：1）xlncosyylnsin2）等式兩邊同時(shí)對x求導(dǎo)數(shù)lncosyxcosdylncosyycot lnsinxxtan1t1tyarctan

11t2

x

x

ln1t2

21t22 2yarctandyyt

yarctan

yt

1t x3t22t9、設(shè)eysinty10：1）t0y

t2)

eyysinteycosty

y

ecos eycost1eycost1eysint6t2

1eysindyyt1eysint

6t

t

t0, 10、設(shè)fxln x1，求f2，f1x x1【解1）f2

f

ln

x22）x1fxlnxfx1xx1f

x1fxx1fx1f1limfxf1limx10

x

x1f1limfxf1limlnx0limx1f1f1f1

xx

x

x1 fx

x x11、設(shè)函數(shù)fxx2 x

x1處可導(dǎo)，求a【解：1）可導(dǎo)必連續(xù)，故limfxlimfxf1

limaxb即ab1b1

x1x2

2）f1f1limfxf1limfxf

x1x1

x2limx limax 2

1

lim a

x

x

x1x1x2

xa b12fx1x23x5f【解1）fx1x23x5fxx123x15fx2x132x13fsinx2sin3x3cos2x4ffsinx2sin3x312sin2x4fx2x36x2fx6x2 2 2 214zxz

2y，求x2y2

2

2y2

sin2

2yx26xsin2

2y

cosx

2yy2xsin2

2yy24x

2yxy

2y4xcosx2y15zeuv,uxy,vsinxy，求zzzuzveuv

xy

euv

sinxy u

v

yeuveuvcosxyycosxyexysinxyzzuzveuv

euv

sinx u

v

xeuveuvcosxyxcosxyexysinxy16z2x3yx2，求z z2x3yx2ex2ln2x3z

x2ln2x3y

2xln2x3y2x3yz3x22x3yx2117zexy2xcos2tyln3t2zecos2tln3t22dzecos2tln3t22

2sin2t6ln3t2

18x2z2y2z2y0zzxy，求z：1）Fxyzx2z2y2z2

F F4yz2 Fx24y2z zF

4yz2 x

x4y

z x24y2z 19、設(shè)方程exyy2sinx2y2yyx 【解1exyy2sinx2y2exyyxy2yycosx2y22x2xcosx2y2y

xexy2y2ycosx2y22：1）Fxyexyy2sinx2y2 Fyexy2xcosx2y2, Fxexy2y2ycosx2y2

yexy2xcosx2y2

2xcosx2y2 x 2ycosxy

xexy2ycosx2y220zzxy由方程exy2zez2所確定，求

y2：1）Fxyzexy2zez2

x2,y1z2Fyexy Fxexy F2

ye x

2xexy