推导过程可参见《古今算学丛书，割圆密率捷法》，清光绪戊戌六月算学书局印成，1898年刘铎整理，

圆周率π=3.141592653589793238462643186367472279514（小于71519），

推导过程参见《古今算学丛书，假数测圆》清光绪戊戌六月算学书局印成，清咸丰壬子年，湖北人戴煦识，夏鸾翔编写，1898年刘铎整理，

以本弧弧分径，求四十五度以内正割对数。

术曰：先求各率分子，为递次乘法，以二为数根，即为第一乘法，置前数根，加二得四，为数根，置前乘法，四五递乘之，一二递除之，得二十，为初减数，以数根减初减数，得十六，为第二乘法，置前数根，加二，得六，为数根，置前初减数，六七递乘之，三四递除之，得七十，为初减数，置前乘法六七，递乘之，一二递除之，得三百三十六，为次减数，以数根减初减数，得六十四，再减次减数，得二百七十二为第三乘法，置前数根加二，得八，为数根，置前初减八九递乘之，五六递除之，得一百六十八，为初减数，置前次减八九递乘之，三四递除之，得二千零十六，为次减数，置前乘法八九递乘之，一二递除之，得九千七百九十二，为三减数，以数根减初减，得一百六十，再减次减，得一千八百五十六，再减三，减得七千九百三十六，为第四乘法，凡数根均起各偶数，其求各减数，则用偶奇二数，乘而逐次，乘法递加，如第二乘法，用四五乘，第三乘法用六七乘，再用奇偶二数，除而，挨次减数递降，如第三乘法，初减用三四除，次减用一二除，乘法将一位，则多一减，如是递求得各率分子，即为递次乘法。

根据以上描述，推导出

第一乘法    二           2                                 S   =2

                                                             1  

                        2*4*5          

第二乘法    一六                   -2*2=16                      S   =16      

                         1*2                                 2        

                          20

                       16*6*7    20*6*7    

第三乘法    二七二                 -         +2*3=272            S   =272      

                         1*2       3*4                         3    

                         336       70    

                       272*8*9   336*8*9  70*8*9  

第四乘法    七九三六               -        +         -2*4=7936   S   =272      

                         1*2       3*4     5*6                  4    

                        9792      2016     168

                   7936*10*11  9792*10*11  2016*10*11  168*10*11

第五乘法  三五三七九二           -            +           -         +2*5= 353792    

                      1*2         3*4         5*6        7*8                

                     436480      89760        7392       330      10

                                                              S    =353792

                                                                5

第六乘法  三五三七九二

353792*12*13   436480*12*13   89760*12*13    7392*12*13   330*12*13

             -              +               -           +          -2*6=22368256  

1*2            3*4             5*6            7*8        9*10        

27595776       5674240        466752         20592       572      12

                                                              S  = 22368256

                                                                6

第七乘法     一九零三七五七三零零

第八乘法     二零九八六五三零零零零零

第九乘法     二九零八八八九零零零零零零零

第十乘法     四九五一五零零零零零零零零零零零

第十一乘法   一零一五四二零零零零零零零零零零零零零

第十二乘法   一零一五四二零零零零零零零零零零零零零

第十三乘法   七零二五二零零零零零零零零零零零零零零零零零零

第十四乘法   二三一二零零零零零零零零零零零零零零零零零零零零零零

把上面的计算过程，用数学归纳法，得到下面的公式

                     S   *(2n-2)(2n-1)             S  *(2n-4)(2n-3)

                       n-2                         n-3              

       S   *2n(2n+1)                 *2n(2n+1)                   *2n(2n+1)            

        n-1                1*2                        1*2

S    =             -                          +                          …-2              

n        1*2                3*4                         5*6

对数的计算, lg0.98=(1-0.98)*0.434294482,

a     对数根     0.434294482                                     a=0.434294482

b     第一数     (1-0.98)*0.434294482=0.00868588964                b=(1-N)*a

c     第二数      0.00868588964*0.02/2=0.00008685890               c=b(1-N)/2

d     第三数      0.00008685890*0.02*2/3=0.00000115812             d=c*(1-N)*2/3

e     第四数      0.00000115812*0.02*3/4=0.00000001737             e=d*.(1-N)*3/4

f      九率        0.000000017378*0.02*4/5=0.00000000028           f=e*(1-N)*4/5

lg0.98=-0.00868588964-0.00008685890-0.00000115812-0.00000001737-0.00000000028=-0.00877392431,

lg98=2-lg0.98=2-0.00877392431=1.99122607569,

当N<1时

                          2     3              4                5        

                     (1-N)   (1-N)    2     (1-N)   2     3    (1-N)   2    3   4

lgN=0.434294482[(1-N)+      +            +                  +                  

                       2      2     3       2    3     4      2    3    4    5    

         n      

     (1-N)    2     5     4       n-1

+…+                          …       ]

      2      3     4     5        n

                  m              

当N>1时，且N/10    <1,

                        m  2        m  3               m  n  

                  (1-N/10  )   (1-N/10  )     2      (1-N/10  )   2    3    4   n-1

lgN=0.434294482[m-[           +                 +..+                        …  ]]  

                       2          2        3          2       3    4    5    n

因为,    可参见高等教育出版社菲赫金哥尔茨著1953年版《微积分教程》第二卷第二分册,

398.对数的计算,

      lgn

ln n=

       lge

所以,

        lgsecθ

lnsecθ=      

lge

e=2.71828182846, lge=0.4342944819,

        lgN

lnN=      

0.4342944819

当N<1时

                          2     3              4                5        

                     (1-N)   (1-N)    2     (1-N)   2     3    (1-N)   2    3   4

lnN= (1-N)+      +            +                  +                  

                       2      2     3       2    3     4      2    3    4    5    

         n      

     (1-N)    2     5     4       n-1

+…+                          …       ]

      2      3     4     5        n

                  m              

当N>1时，且N/10    <1,

                        m  2        m  3               m  n  

                  (1-N/10  )   (1-N/10  )     2      (1-N/10  )   2    3    4   n-1

lnN=m-[           +                 +..+                        …    ]  

                       2          2        3          2       3    4    5    n

正割对数的计算公式

法检弧线表，得四十五度，弧分单位下，七八五三九八一六三四零为二率，自乘，得单位下六一六八五零二七五零七二，为三率，以对数根，单位下四三四二九四四八一九零三乘之，二除之，得零一三三九四七三三五三一，为第一数正，次置第一数，以三率乘之，得五率，三除之，四除之，得连单位三零下六八八五四五四二一九二六，为七率，用数第一乘法，二乘之，得一三七七零九零八四四，为第二数正，次置七率，用数以三率乘之，得七七六三八，为九率，用数第二乘法，一六乘之，得二二六五二二三六四，为第三数正，次置九率，用数以三率乘之，得九率，七除之，八除之，得连单位六零下一五五九四九零八七八二，为十一率，用数第三乘法二七二乘之，得四二四一八一五二，为第四数正，次置十一率，用数以三率乘之，得十一率，九除之，十除之，得连单位八零下一零六八八五八一九七，为十三率，用数第四乘法七九三六乘之，得八四八二四五九，为第五数正，次置十三率，用数以三率乘之，得十三率，十一除之，十二除之，得连单位十一零下四九九四八八九九五，为十五率，用数第五乘法三五三七九二乘之，得一七六七一五二，为第六数正，次置十五率，用数以三率乘之，得十五率十三除之，十四除之，得连单位十三零下一六九二九一一七，为十七率，用数第六乘法二二三六八二五六乘之，得三七八六七五，为第七数正。次置十七率，用数以三率乘之，得十七率，十五除之，十六除之，得连单位十六零下四三五一一三七七，为十九率，用数第七乘法一九零三七五七三下连单位二零乘之，得八二八三五，为第八数正，次置十九率用数以三率乘之，得十九率，十七除之，十八除之，得连单位十九零下八七七一二四三，为二十一率，用数第八乘法二零九八六五三下，连单位五零乘之，得一八四零八，为第九数正，次置二十一率，用数以三率乘之，得一八四零八，为第九数正，次置二十一率，用数以三率乘之，得二十一率，十九除之，二十除之，得连单位二十一零下一四二三八二七，为二十三率，用数第九乘法二九零八八八九下，连单位七零乘之，得四一四二，为第十数正，次置二十三率，用数以三率乘之，得二十三率，二十一除之，二十二除之，得连单位二十四零下一九零一零五，为二十五率，用数第十乘法四九五一五零下，连单位十零，乘之，得九四一，为第十一数正，次置二十五率，用数以三率乘之，得二十五率，二十三除之，二十四除之，得连单位二十七零下二一二四四，为二十七率，用数第十一乘法一零一五四二下，连单位十三零乘之，得二一六，为第十二数正，次置二十七率，用数以三率乘之，得二十七率，二十五除之二十六，除之，得连单位三十零下二零一六零，为二十九率，用数第十二乘法二四六九二下连单位十六零，乘之，得五零，为第十三数正，次置二十九率，用数以三率乘之，得二十九率，二十七除之，二十八除之，得连单位三十三零下一六四五，为三十一率，用数第十三乘法，七零二五二下连单位十八零乘之，得一十二，为十四数正，次置三十一率，用数以三率乘之，得三十一率，二十九除之，三十除之，得连单位三十六零下一一七第十四，乘法二三一二下连单位二十一零乘之，得三，为第十五数正，乃以诸正数相并，得零一五零五一四九九七八四，以半径一百亿系十一位乃于首位加一零，尾位未满五弃之，得一零一五零五一四九九七八，为四十五度正割对数也。

余切对数求法

lgsec44°+10=10.1430659099,

lgsec44°+20=20.1430659099,

lgcsc44°+10=10.1582287268,

lgtg44°=lgsec44°+20-lgcsc44°-10-10=20.1430659099-10.1582287268-10=9.9848371831-10=-0.015162817,

lgctg44°=lgcsc44°+20-lgsec44°-10-10=20.1582287268-10.1430659099-10=0.01516282,

lgctgθ=lgcscθ-lgsecθ,

lgctg44°=lgcsc44°-lgsec44°=-0.1431+0.1582=0.0151,

lgtgθ=lgsecθ-lgcscθ,

lgctgθ=lgcscθ-lgsecθ,

lntgθ=lnsecθ-lncscθ,

lnctgθ=lncscθ-lnsecθ,

正弦对数求法

lgcsc44°+10=10.1582287269,

lgsin44°=20-lgcsc44°-10-10=9.8417712731-10=-0.158287269,

lgsinθ=-lgcscθ,

lgcosθ=-lgsecθ,

lnsinθ=-lncscθ,

lncosθ=-lnsecθ,

正矢对数求法

44°/2=22°,

45°-22°=23°,

lg2+3=3.03010299956,

2*[(lgcsc22°)/10+1]=2*1.04264245830=2.08528491660,

lgversin44°=[lg2+3-2*[(lgcsc22°)/10+1]-1]*10=(3.03010299956-2.08528491660-1)*10=(0.94481808296-1)*10=-0.5518190172,

lgversin44°=lg(1-cos44°)=lg0.28066=-0.551819479,

lgversinθ=[lg2+3-2*[lgcsc(θ/2)]/10+1]-1]*10,

2*[(lgcsc23°)/2+1]=1.04081219884*2=2.08162439768,

lgvercos44°=2*[(lgcsc23°)/2+1]-[lg2+3-2*[(lgcsc22°)/10+1]]*10-10=2.08162439768-9.4481808296-10=9.4847860188-10=-0.515213982,

lgvercosθ=2*[(lgcsc(90°-θ))/2+1]-[lg2+3-2*[(lgcscθ/2)/10+1]]*10-10,

正大矢对数求法

44°/2=22°,

45°-22°=23°,

lg2+3=3.03010299956,

2*[(lgcsc22°)/10+1]=2*1.00328341395=2.00656682790,

lgvercos23°=10*[lg2+3-2*[(lgcsc22°)/10+1]-10]-1=[3.03010299956-2.00656662790-10]*10-1=1.02353617166*10-1=0.2353617166

lgvercosθ=10*[lg2+3-2*[(lgcsc(45°-θ)/10)+1]-10]-1

2*[(lgsec23°)/10+1]=1.00359739173*2=2.00719478346,

lgvercos22°=lg2+3-2*[(lgcsc23°)/10+1]-10=3.03010299956-2.00719478346-10=1.02290821610

正割对数计算公式

对数根0.434294481903，

a    二率    θ=0.78539816340                                    a=θ

             2                                                        2  

b     三率 θ =0.78539816340*0.78539816340=0.616850275072          b=θ

c  第一数正 0.616850275072*0.434294481903/2=0.13394733531    c=0.434294481903*b/2

d 七率  0.13394733531*0.616850275072/3*4=0.00688545421926    d=c*b/3*4

e第二数正   0.00688545421926*2=0.01377090844               e=2d

f    九率  0.00688545421926*0.616850275072/5*6=0.00014157648  f=bd/5*6

g第三数正0.00014157648*16=0.00226522364                    g=f*16

h第十一率0.00014157648*0.616850275072/7*8=0.00000155949     h=bf/7*8

i第四数正0.00000155949*272=0.0004218153                     i=272h

j第十三率0.00000155949*0.616850275072/9*10=0.00000001069     j=bh/9*10

k第五数正0.00000001069*7936=0.00008482454                  k=7936*j

m第十五率0.00000001069*0.616850275072/11*12=0.0000000000499488995  m=bj/11812

n第六数正0.0000000000499488995*353792=0.00001767152        n=353792m

o第十七率0.0000000000499488995*0.616850275072/13*14=169291167E13  o=bm/13*14

p第七数正169291167E13*22368256=378674816E6                 p=22368256o

q第十九率169291167E13*0.616850275072/15*16=43511377E15     q=bo/15*16

s第八数正43511377E15*1903757300=8283E5                    s=1903757300q

t第二十一率43511377E15*0.616850275072/17*18=877124343196E18   t=bq/17*18

u第九数正877124343196E18*209865300000=184077963212E6    u=209865300000t

当45°≥θ>0°时

                      2                 2   2                                                

       0.434294481903θ   0.434294481903θ   θ     2    

lgsecθ=                +                             +  

            2                  2           1    3*4                  

                      2    2         2                            2   S  

       0.434294481903θ   θ     2    θ     16       0.434294481903θ      n

                                     +...+

            2            1    3*4   1     5*6             2      (n+1)(n+2)...*2n                            

                          2      4            6                8

                         θ     θ     2      θ    2    16     θ     2   16 272

lgsecθ=0.434294481903（     +           +                 +  

                          2     2   3*4     2    3*4   5*6    2  3*4  5*6  7*8

      10                            2n     S              

    θ     2   16   272  7936      θ          n

+                           +…+                    ）    

     2  3*4  5*6  7*8   9*10       2   (n+1)(n+2)...*2n

上式中，

                          S     *(2n-2)(2n-1)                S     *(2n-2)(2n-1)          

                            n-2                              n-3

       S    *2n(2n+1)                          *2n(2n+1)                *2n(2n+1)            

         n-1                     1*2        

S   =                  -                                 +                  …-2              

n        1*2                          3*4                           5*6        

lgsec45°=lgsec0.78539816340=0.13394733531+0.01377090844+0.00226522364+0.0004218153+0.00008482454=0.15049010723，

当67.5°≥θ>45°时

lgsecθ=lgsec(2θ-90°)-lgsec(90°-θ)+lg2,

  当78.75°>θ≥67.5°时

lgsecθ=lgsec[2(2θ-90°)-90°]-lgsec2(90°-θ)-lgsec(90°-θ)+2lg2,

当84.375°>θ≥78.75°时

lgsecθ=lgsec[2[2(2θ-90°)-90°]-90°]-lgsec4(90°-θ)-lgsec2(90°-θ)-lgsec(90°-θ)+3lg2,

当85.375°>θ≥84.375°时

lgsecθ=lgsec[2[2[2(2θ-90°)-90°]-90°]-90°]-lgsec6(90°-θ)-lgsec4(90°-θ)-lgsec2(90°-θ)-lgsec(90°-θ)+4lg2,

当86.375°>θ≥85.375°时,

lgsecθ=lgsec[2[2[2[2(2θ-90°)-90°]-90°]-90°]-90°]-lgsec8(90°-θ)-lgsec6(90°-θ)-lgsec4(90°-θ)-lgsec2(90°-θ)-lgsec(90°-θ)+5lg2,

当87.375°>θ≥86.375°时,

lgsecθ=lgsec[2[2[2[2[2(2θ-90°)-90°]-90°]-90°]-90°]-90°]-lgsec8(90°-θ)-lgsec8(90°-θ)-lgsec6(90°-θ)-lgsec4(90°-θ)-lgsec2(90°-θ)-lgsec(90°-θ)+6lg2,

当88.375°>θ≥87.375°时,

lgsecθ=lgsec[2[2[2[2[2[2(2θ-90°)-90°]-90°]-90°]-90°]-90°]-90°]-lgsec12(90°-θ)-lgsec10(90°-θ)-lgsec8(90°-θ)-lgsec6(90°-θ)-lgsec4(90°-θ)-lgsec2(90°-θ)-lgsec(90°-θ)+7lg2,

因为, 可参见高等教育出版社菲赫金哥尔茨著1953年版《微积分教程》第二卷第二分册,

398.对数的计算,

     lgn      

ln n=

      lge

所以,

        lgsecθ      

lnsecθ=

          lge

e=2.71828182846, lge=0.4342944819,

            lgsecθ      

lnsecθ=

         0.4342944819

所以，当45°≥θ>0°时

          2       4              6                   8    

        θ       θ      2       θ     2      16     θ      2    16     272

lnsecθ=       +            +                     +  

        2        2    3*4      2    3*4     5*6     2    3*4   5*6    7*8

          10                                  2n       S      

        θ      2    16    272   7936        θ             n  

+                                 +…+              

        2      3*4  5*6   7*8   9*10        2      (n+1)(n+2)...*2n

上式中，

                          S     *(2n-2)(2n-1)                S     *(2n-2)(2n-1)          

                            n-2                              n-3

       S    *2n(2n+1)                          *2n(2n+1)                *2n(2n+1)            

         n-1                     1*2        

S   =                  -                                 +                  …-2              

n        1*2                          3*4                           5*6    

当67.5°≥θ>45°时

lnsecθ=lnsec(2θ-90°)-lnsec(90°-θ)+ln2,

  当78.75°>θ≥67.5°时

lnsecθ=lnsec[2(2θ-90°)-90°]-lnsec2(90°-θ)-lnsec(90°-θ)+2ln2,

当84.375°>θ≥78.75°时

lnsecθ=lnsec[2[2(2θ-90°)-90°]-90°]-lnsec4(90°-θ)-lnsec2(90°-θ)-lnsec(90°-θ)+3ln2,

当85.375°>θ≥84.375°时

lnsecθ=lnsec[2[2[2(2θ-90°)-90°]-90°]-90°]-lnsec6(90°-θ)-lnsec4(90°-θ)-lnsec2(90°-θ)-lnsec(90°-θ)+4ln2,

当86.375°>θ≥85.375°时,

lnsecθ=lnsec[2[2[2[2(2θ-90°)-90°]-90°]-90°]-90°]-lnsec8(90°-θ)-lnsec6(90°-θ)-lnsec4(90°-θ)-lnsec2(90°-θ)-lnsec(90°-θ)+5ln2,

当87.375°>θ≥86.375°时,

lnsecθ=lnsec[2[2[2[2[2(2θ-90°)-90°]-90°]-90°]-90°]-90°]-lnsec8(90°-θ)-lnsec8(90°-θ)-lnsec6(90°-θ)-lnsec4(90°-θ)-lnsec2(90°-θ)-lnsec(90°-θ)+6ln2,

当88.375°>θ≥87.375°时,

lnsecθ=lnsec[2[2[2[2[2[2(2θ-90°)-90°]-90°]-90°]-90°]-90°]-90°]-lnsec12(90°-θ)-lnsec10(90°-θ)-lnsec8(90°-θ)-lnsec6(90°-θ)-lnsec4(90°-θ)-lnsec2(90°-θ)-lnsec(90°-θ)+7ln2,

例如：

lgsec2°+10=10.0002646411，

lgsec2°+20=20.0002646411，

lgsec44°+10=10.1430659099，

lg2=0.301029995，

lgsec44°+10-lg2=10.1430659099-0.301029995=9.842035904，

lgsec46°=lgsec2°+10+10-lgsec44°-10-lg2-10=20.0002646411-9.842035904-10=10.15822874-10=0.15822874，

lnsecθ=lnsec(2θ-90°)-lnsec(90°-θ)+lg2，

例如：

lgsec4°+10=10.00010592102，

lgsec4°+20=20.00010592102，

lgsec43°+10=10.1358725362，

lg2=0.301029995，

lgsec43°+10-lg2=10.1358725362-0.301029995=9.8348425406，

lgsec47°=lgsec4°+10+10-lgsec43°-10+lg2-10=20.00010592102-9.8348425406-10=10.1662166696-10=0.1662166696，

lnsecθ=lnsec(2θ-90°)-lnsec(90°-θ)+lg2

0.00011-0.1359+0.301029995

lnsecθ=lnsec(2θ-90°)-lnsec(90°-θ)+lg2，

例如：

lnsec67°=lnsec(2*67°-90°)-lnsec(90°-67°)-lg2

       =lnsec44°-lnsec23°+lg2

因为，

lnsinθ=-lncscθ，

lncosθ=-lnsecθ，

lnsec67°=-lncos44°+lncos23°+lg2

    =0.1431-0.036+0.301029995=-0.408129995

例如：

lgsec68°=lgsec(2*68°-90°)-lg(90°-68°)+lg2

       =lgsec46°-lgsec22°+lg2

lgsec68°=lgsec(2*46°-90°)-lgsec(90°-46°)+lg2-lgsec22°+lg2

       =lgsec2°-lgsec44°+lg2-lgsec22°+lg2

       =0.30003-0.1431-0.0328+0.301029995

       =0.42519995

例如：

lgsec79°=lgsec(2*79°-90°)-lg(90°-79°)+lg2

       =lgsec68°-lgsec11°+lg2

lgsec68°=lgsec(2*46°-90°)-lgsec(90°-46°)+lg2-lgsec22°+lg2

       =lgsec2°-lgsec44°+lg2-lgsec22°+lg2

       =0.30003-0.1431-0.0328+0.301029995

       =0.42519995

lgsec79°=lgsec(2*79°-90°)-lg(90°-79°)+lg2

      =lgsec68°-lgsec11°+lg2

      =0.42519995-0.00081+0.301029995

.      =0.725419495

例如：

lgsec85°=lgsec(2*85°-90°)-lg(90°-85°)+lg2

       =lgsec80°-lgsec5°+lg2

lgsec80°=lgsec(2*80°-90°)-lg(90°-80°)+lg2

        =lgsec70°-lgsec10°+lg2

lgsec70°=lgsec(2*70°-90°)-lg(90°-70°)+lg2

        =lgsec50°-lgsec20°+lg2

lgsec50°=lgsec(2*50°-90°)-lg(90°-50°)+lg2

        =lgsec10°-lgsec40°+lg2  

  推导过程可参见《对数表新编》冯度编开明书店出版1935年版

     logcosα计算公式，当88°<α<90°时，

如果88°≤α<90°，根据《对数表新编》中的S,T公式，判断余弦对数值，

log cosα=log (90°-α)``+S，

log cotα=log (90°-α)``+T，

上式中，

90°-α=MN°WS`T``，

(90°-α)``=3600*MN+60*WS+T，

log (90°-α``)/1000=lgA.BC，

如果0≤(90°-α``)<7267，

那么，log cosα=log (90°-α``)/1000+3+4.68553，

              =lg A.BC+3+4.68553，

计算log cosα时，首先计算log(90°-α``)/1000,再加上3，最后加上4.68553，这样得到的数后面附加-10，给这个数减去10，就是log sinα的值，

例如:

log cos88°26`41.2``=log5598.8+4.68553≈3.74809+4.68553 - 10 ≈8.43362 - 10 ≈-1.56639，

log cos88°26`41.2``=log5591.87+4.68553≈3.7462+4.68553 - 10 ≈8.43173 - 10 ≈-1.56827，

0.027145*10000=271.45*20.6=5591.87，

90°-88°26`41.2``=1°33`18.8``=0.027145，

第四部分导数的定义

推导过程可参见1992年版《高等数学》，盛祥耀主编，高等教育出版社出版

三   导数的几何意义

根据导数定义及曲线的切线的斜率的求法，我们可以知道，

函数y=f(x)在点x    处的导数的几何意义就是曲线y=f(x)在点(x   ,f(x    ))处的切线的斜率，如图2-2，即

tga=f`(x    )

        0

由此可知曲线y=f(x)上点P   处的切线方程为

                        0

   y-y    =f`(x    )(x-x     )

      0       0       0

法线方程为

            -1

y-y    =           (x-x   )(f`(x  )≠0）  

    0     f`(x   )       0    0

              0        

积分表        

kdx=kx+C  

       μ     1     μ-1

x  dx=        x    +C    (μ≠-1)

             μ+1                      

dx/x=ln│x│+C  

        x      x              

      a   dx=a   /lna+C

当a=e时，

        x      x              

      e   dx=e    +C

      cosxdx=sinx   +C

      sinxdx=-cosx   +C

          2        

      sec   xdx=tgx    +C

          2        

      csc   xdx=-ctgx    +C

      secxtgxdx=secx    +C

      cscxctgxdx=-cscx   +C

        dx          

               =arcsinx+C=-arccosx     +C

          2

1-x

        dx          

               =arctgx+C=-arcctgx   +C

          2

1-x

      shxdx=chx   +C

       chxdx=shx  +C

        m      m+1        

      x    dx=x     /(m+1)+C

      dx/x=         d(-x)/(-x)=log│x│+c

        x      x      

      a    dx=a    /log a    +c

       cosxdx=sinx   +C

      sinxdx=-cosx  +C

            2          

      dx/cos   x=tan x    +c

                 2       ±arc sinx+c    

      dx/     1-x     ={

                         ±arc cosx+c

            2          

      dx/ (x   +1)  =arc tanx+c

      chxdx=shx+c

      shxdx=chx+c

           2      

      dx/ch   x=thx+c

               2      

      dx/     x  -1   =±argchx+c

             2          

       dx/(1-x   )=±argthx+c

积分计算过程

根据下面的公式，tga=y`=f`(x)=u(x)=y/x，a=arctgy`，

            3        5                 2n+1

          f`  (x)   f`   (x)          n  f`   (x)       2n+2

a=f`(x)-        +         -...+(-1)             +o(x      )    

           3        5                2n+1          

                     2        5        6               2m

                    a        a       a           m   a         2m+1

f(x)=-lncosa+C=-ln[1-       +        -         -...+(-1)          +o(a      )]+C    

                   2!        4!      6!             (2m)!  

                             1         2   1         3       6

f(x)=lnseca+C=(sec a-1)-     (sec x-1) +      (sec x-1)  +o(a   )

                              2            3

                      2        4        6          

                    a        a       a          6

f(x)=-lncosa+C=       +        +         +o(a      )

                2        12     45      

kdx=kx+C  

                      2        4        6          

                    a        a       a          6

f(x)=-lncosa+C=       +        +         +o(a      )

                2        12     45      

            3        5               2n+1

           x        x           n   x          2n+2

arc tg x=x-       +         -...+(-1)          +o(x      )    

       3        5              2n+1        

a=arctgy`,上式中,

            3        5               2n+1

           k        k           n   k          2n+2

a=k-       +         -...+(-1)          +o(x      )    

       3        5              2n+1      

                             3       5               2n+1  

                    1       k      k         n     k             2n+2    

f(x)=-lncosa+C=       [k-     +       -…+(-1)                +o(a      ) ]

                2       3      5              2n+1      

                             3       5               2n+1  

                    1       k      k         n     k             2n+2    

+       [k-     +       -…+(-1)                +o(a      ) ]

                12      3      5              2n+1      

                             3       5               2n+1  

                    1       k      k         n     k             2n+2    

+       [k-     +       -…+(-1)                +o(a      ) ]

                45      3      5              2n+1      

           =kx

         2  

csc   xdx=-ctgx+C=f(x)

                      2        4        6          

                    a        a       a          6

f(x)=-lncosa+C=       +        +         +o(a      )

                2        12     45      

            3        5               2n+1

           x        x           n   x          2n+2

arc tg x=x-       +         -...+(-1)          +o(x      )    

           3        5              2n+1  

   a=arctgy`,上式中,

               2   3       2  5            2   2n+1

     2     (csx   x)    (csx  x)        n  (csx  x)         2n+2

a=(csx   x)-         +          -...+(-1)              +o(x      )    

          3         5                 2n+1      

                                2   3    2   5            2  2n+1  

                1       2    (csx  x)  (csx  x)       n  (csx  x)           2n+2    

f(x)=-lncosa+C=       [(csx  x)-        +        -…+(-1)                +o(a      ) ]

           2                3      5               2n+1      

                                 2  3    2  5            2  2n+1  

                1       2    (csx  x)  (csx  x)       n  (csx  x)           2n+2    

+      [(csx  x)-        +         -…+(-1)                +o(a      ) ]

           12               3       5               2n+1      

                                 2  3    2  5            2  2n+1  

                1       2    (csx  x)  (csx  x)       n  (csx  x)           2n+2    

+      [(csx  x)-        +         -…+(-1)                +o(a      ) ]

           45               3       5               2n+1      

       =-ctgx

shxdx=chx   +C

                      2        4        6          

                    a        a       a          6

f(x)=-lncosa+C=       +        +         +o(a      )

                2        12     45    

   上式中

               3        5            2   2n+1

            sh   x     sh   x       n     sh   x        2n+2

a=sh x-           +          -...+(-1)              +o(x      )    

          3         5                 2n+1      

                            3        5               2n+1            

                1        sh   x    sh  x        n    sh   x          2n+2    

f(x)=-lncosa+C=       [shx-        +         -…+(-1)                +o(a      ) ]

           2           3        5               2n+1      

                            3        5               2n+1            

                1        sh   x    sh  x        n    sh   x          2n+2    

+       [shx-        +         -…+(-1)                +o(a      ) ]

           12           3        5               2n+1    

                            3        5               2n+1            

                1        sh   x    sh  x        n    sh   x          2n+2    

+       [shx-        +         -…+(-1)                +o(a      ) ]

           45           3        5               2n+1      

       =chx

例1.

         3    2        

(4x   -2x   -5x-3)dx

              3               2                              

=4        x   dx-         2x   dx+         5xdx-        3dx        

     4        3        2

     x        x        x  

=4       -2       +5       -3x+C        

4        3        2

     3       2      

     2x      5x      

=x-       +       -3x+C        

3        3      

         3    2        

(4x   -2x   -5x-3)dx

                      2        4        6          

                    a        a       a          6

f(x)=-lncosa+C=       +        +         +o(a      )

                2        12     45    

上式中      

                     3    2     3     3   2      5          3    2      2n+1                              

     3     2      (4x  -2x  -5x-3)   (4x  -2x  -5x-3)      n  (4x  -2x  -5x-3)    2n+2    

a=(4x   -2x  -5x-3)-               +               -…+(-1)                 +o(a  ) ]

                    3              5                      2n+1      

导数公式表

(c)`=0,

  a     a-1              

(x  )`=ax

  x     x          

(a   )`=a   lna

   x    x      

(e    )`=e

(log  x)`=1/xlna

   a

(lnx)`=1/x

(sinx)`=cosx

(cos)`=-sinx

         2    

(tgx)`=sec   x

         2

(ctgx)`=-csc   x

(secx)`=secxtgx,

(cscx)`=-cscxctgx,

             1  

(arcsinx)`=        2

             1-x

             -1  

(arccosx)`=        2

             1-x

             1  

(arctgx)`=        2

             1+x

             -1  

(arcctgx)`=        2

             1+x

  x      x      x                                

(a    )`=a   lga =a  /log  e

                      a

(log   x)`=log  e/x=1/(xlog a)

a       a

(lg  x)`=1/x

                      2      

   (arc cosx)`=-ε/    1-x          ε=±1,其号与siny之号同

                      2      

   (arc sinx)`=ε/    1-x          ε=±1,其号与cosy之号同

               2      

(arc tanx)`=1/ (1+x   )

(u+v+w)`=u`+v`+w`   (u,v,w,表x之函数而有引数u`,v`,w`者

(uvw)`=u`/u+v`/v+w`/w  

                2    

(u/v)`=[(vu`-uv`)/v   ]

  v    v          v-1                    

u   =u   v`log u+vu  u`

x=φ(y)  [x=φ(y)表y=f(x)之反函数，而y有引数f`(x)=0]

x`=φ`(y) =1/f`(x)

[备考]——三角函数cotx=cosx/sinx,secx=1/cosx,cscx=1/sinx,versx=1-cosx,covsx=1-sinx等或为两函数之商，或为两函数之和，

导数计算过程

因为，tga=y`=f`(x)=u(x)，

                        3              

                   arctg   (y/x)      

y`=u(x)=t=arctg(y/x)+

                        3

  a              

(x   )`=ax,

                          3   a            

               a     arctg   (x   /x)      a-1

y`=u(x)=t=arctg(x   /x)+                  =ax  

                         3

(log   x)`=1/xlna

    a

                         3      

                     arctg   (log   x/x)            

                                a            

y`=arctg(log    x/x)+                     =1/xlna

           a              3

              4     x                

例3.设f(x)=3x     -e     +5cosx-1,求f`(x)及f`(0)

                     4       4                

解:根据理论1可得(3x   )`=3(x   )`,(5cosx)`=5(cosx)`,  

又,

   4      3                x      x                              

(x    )`=4x    ,(cosx)`=-sinx,(e    )`=e    (1)`=0,

故,

        4     x                        

f`(x)=(3x    -e   +5cosx-1)`

     4      x                

=(3x    )`-(e   )+(5cosx)`-(1)`

     3     x    

=12x    -e    -5sinx

         3   x            

f`(0)=(12x   -e   -5sinx)    =-1

                         x=0

                                      4  x                        

           4   x              arctg[(3x  -e  +5cosx-1)/x]        3  x

y`=arctg[(3x   -e  +5cosx-1)/x]+                             =12x  -e  -5sinx

                                         3