高數(shù)重要公式

1、重要極限公式：推廣式：ennn????)11(limenfnfn????)())(11(lim????)(nfn推廣式：1lim0??xSinxx1)()inf(lim0??xfxSx重要等價無窮小公式（）：sinxxtanxxarcsinxxarctanxx0?x????ln(1x)xx1cosx?1?xe??221xxx?????1)1(導數(shù)公式：(C)=0()=()=()=()=(lnx)=?x1???xxaaaxlnxexexa

2、logaxln1x1(sinx)=cosx(cosx)=sinx(tanx)=sec2x(cotx)=csc2x(arcsinx)=(arccosx)=(arctanx)=(arccotx)=211x?211x??211x?211x??導數(shù)定義式：或ooxoxoxxxfxfxf????)()(lim)(hxfhxfxfh)()(lim)(0????微分公式：d(c)=0dxaxxdaa1)(??dxaxxdaln1)(log?dxxxd

3、1)(ln?adxaadaxln)(?dxeedxx?)(xdxxdcos)(sin?xdxxdsin)(cos??xdxdxxxd22seccos1)(tan??xdxxxdtansec)(sec?xdxxxdcotcsc)(csc??dxxxd211)(arcsin??dxxxd211)(arccos???dxxxd211)(arctan??dxxxarcd211)cot(???中值定理：如果函數(shù)f(x)滿足條件1.在閉區(qū)間[ab]

4、上連續(xù)2.在開區(qū)間（ab）內(nèi)可導3.f(a)=f(b)則至少存在一點ξ，使得)(ba??0)(??f拉格朗日中值定理：如果函數(shù)f(x)滿足條件1.在閉區(qū)間[ab]上連續(xù)2.在開區(qū)間（ab）內(nèi)可導則至少存在一點ξ，使得)(ba??abafbff???)()()(?柯西中值定理：f(x)F(x)滿足1.[ab]連續(xù)2.在（ab）可導且F’(x)≠0則（ab）上至少存在一點ξ：)()()()()()(??FfaFbFafbf???曲線在點的切

5、線方程：)(xfy?ox))((ooxxxfyy???曲線在點的法線方程：)(xfy?ox)()(1ooxxxfyy????曲線的凹凸性和拐點：設(shè)y=f(x)在（ab）內(nèi)二階可導1.如果在（ab）內(nèi)的每一點x，恒有f”(x)0則曲線y=f(x)在（ab）內(nèi)是凹的；2.如果在（ab）內(nèi)的每一點x，恒有f”(x)《0則曲線y=f(x)在（ab）內(nèi)是凸的。3.y=f(x)在xo處二階可導，（xof(xo)）為拐點f”(xo)=0隱函數(shù)的導數(shù)：

6、若則其隱函數(shù)為，其導數(shù)為0)(?yxF)(xfy?)()(yxFyxFyyx??隱函數(shù)的偏導數(shù)：由于方程確定隱函數(shù)偏導數(shù)0)(?zyxF)(yxfz?)()(zyxFzyxFxzzx???)()(zyxFzyxFyzzy???二重積分的性質(zhì)：?????DDdyxfkdyxkf??)()(???????????DDDdyxgdyxfdyxgyxf???)()()()(????????21)()()(DDDdyxfdyxfdyxf???如果

7、在D上有σ為D的面積則1)(?yxf???????DDdd???1三角函數(shù)：222212211cos12sinududxxtguuuxuux????????，，，bababasincoscossin)sin(???bababasinsincoscos)cos(???bababatantan1tantan)tan(????bababacotcot1cotcot)cot(????2cos2sin2sinsinbababa????2sin2c

8、os2sinsinbababa????2cos2cos2coscosbababa????2sin2sin2coscosbababa?????bababacoscos)sin(tantan???baabbasinsin)sin(cotcot???)sin(21)sin(21cossinbababa????)cos(21)cos(21sinsinbababa????)cos(21)coscos(21coscosbababa????aaaco

9、ssin22sin?aaa22sincos2cos??aaa2tan1tan22tan??aaacot21cot2cot2??2cos12sinaa??2cos12cosaa??aaacos1sin2tan??aaacos1sin2cot??函數(shù)角Asincostgctgαsinαcosαtgαctgα90αcosαsinαctgαtgα90αcosαsinαctgαtgα180αsinαcosαtgαctgα180αsinαcosαt