导数与微分中值定理口诀

导数与微分中值定理口诀导数与微分中值定理口诀口诀（18）：切线斜率是导数，法线斜率负倒数。口诀（19）：可导可微互等价；它们都比连续强。口诀（20）：有理函数要运算；最简分式要先行。口诀（21）：高次三角要运算；降次处理先开路。口诀（22）：导数为零欲论证；罗尔定理负重任。口诀（23）：函数之差化导数；拉氏定理显神通。口诀（24）：导数函数合为零；辅助函数用罗尔。口诀（25）：寻找sn无约束，柯西拉氏先后上。口诀（26）：寻找sn有约束，两个区间用拉氏。口诀（27）端点、驻点、非导点，函数值中定最值。口诀（28）凸凹切线在上，下；凸凹转化在拐点。口诀（29）数字不等式难证，函数不等式先行第一部分 三角函数公式·两角和与差的三角函数cos(α+β)=cosα·cosβ-sinα·sinβcos(α-β)=cosα·cosβ+sinα·sinβsin(α±β)=sinα·cosβ±cosα·sinβtan(α+β)=(tanα+tanβ)/(1-tanα·tanβ)tan(α-β)=(tanα-tanβ)/(1+tanα·tanβ)·和差化积公式：sinα+sinβ=2sin[(α+β)/2]cos[(α-β)/2]sinα-sinβ=2cos[(α+β)/2]sin[(α-β)/2]cosα+cosβ=2cos[(α+β)/2]cos[(α-β)/2]cosα-cosβ=-2sin[(α+β)/2]sin[(α-β)/2]·积化和差公式：sinα·cosβ=(1/2)[sin( α+β)+sin(α-β)]coscos

·sinβ=(1/2)[sin(·cosβ=(1/2)[cos(

α+β)-sin(α-β)]α+β)+cos(α-β)]sin

·sin

β=-(1/2)[cos(

α+

β)-cos(

α-β)]·：sin(2α)=2sin·cosα=2/(tan α+cotα)cos(2α)=(cosα)^2-(sin α)^2=2(cos α)^2-1=1-2(sin α)^2tan(2α)=2tan α/(1-tan^2 α)cot(2α)=(cot^2 α-1)/(2cot α)sec(2α)=sec^2 α/(1-tan^2 α)csc(2α)=1/2*sec ·cscα·sin(3

α)=3sin

α-4sin^3

α

=4sin

·sin(60

°+α)sin(60

°-α)cos(3

α)=4cos^3

α-3cos

α

=4cos

·cos(60

°+α)cos(60

°-α)tan(3

α)=

(3tan

α-tan^3

α)/(1-3tan^2

α)=

tan

αtan(

π/3+α)tan(π/3-α)cot(3α)=(cot^3 α-3cotα)/(3cot^2 α-1)·n

倍角公式：sin(n α)=ncos^(n-1)+C(n,5)cos^(n-5) ·sin^5cos(n α)=cos^n+C(n,4)cos^(n-4) ·sin^4

·sinα-C(n,3)cos^(n-3)α-⋯α-C(n,2)cos^(n-2)α-⋯

·sin^3α·sin^2

αα·：sin(cos(

α/2)=α/2)=

±√((1-cos±√((1+cos

α)/2)α)/2)tan(

α/2)=

±√((1-cos

α)/(1+cos

α))=sin

α/(1+cos

α)=(1-cosα)/sinαcot(α/2)=±√((1+cos α)/(1-cos α))=(1+cos α)/sinα=sinα/(1-cos α)sec(α/2)=±√((2secα/(secα+1))csc(α/2)=±√((2secα/(secα-1))·辅助角公式：Asinα+Bcosα=√(A^2+B^2)sin(α+φ）（tanφ=B/A）Asinα+Bcosα=√(A^2+B^2)cos(α-φ）（tanφ=A/B）·万能公式sin(a)=(2tan(a/2))/(1+tan^2(a/2))cos(a)=(1-tan^2(a/2))/(1+tan^2(a/2))tan(a)=(2tan(a/2))/(1-tan^2(a/2))·降幂公式sin^2α=(1-cos(2α))/2=versin(2α)/2cos^2α=(1+cos(2α))/2=covers(2α)/2tan^2α=(1-cos(2α))/(1+cos(2α))·三角和的三角函数：sin(α+β+γ)=sinα·cosβ·cosγ+cosα·sinβ·cosγ+cosα·cosβ·sinγ-sinα·sinβ·sinγcos(α+β+γ)=cosα·cosβ·cosγ-cosα·sinβ·sinγ-sinα·cosβ·sinγ-sinα·sinβ·cosγtan(α+β+γ)=(tanα+tanβ+tanγ-tanα·tanβ·tanγ)/(1-tanα·tanβ-tanβ·tanγ-tanγ·tanα)·其它公式·两角和与差的三角函数cos(α+β)=cosα·cosβ-sinα·sinβcos(α-β)=cosα·cosβ+sinα·sinβsin(α±β)=sinα·cosβ±cosα·sinβtan(α+β)=(tanα+tanβ)/(1-tanα·tanβ)tan(α-β)=(tanα-tanβ)/(1+tanα·tanβ)·和差化积公式：sinα+sinβ=2sin[(α+β)/2]cos[(α-β)/2]sinα-sinβ=2cos[(α+β)/2]sin[(α-β)/2]cosα+cosβ=2cos[(α+β)/2]cos[(α-β)/2]cosα-cosβ=-2sin[(α+β)/2]sin[(α-β)/2]·积化和差公式：sinα·cosβ=(1/2)[sin(α+β)+sin(α-β)]cosα·sinβ=(1/2)[sin(α+β)-sin(α-β)]cosα·cosβ=(1/2)[cos(α+β)+cos(α-β)]sinα·sinβ=-(1/2)[cos(α+β)-cos(α-β)]·：sin(2α)=2sin·cosα=2/(tan α+cotα)cos(2α)=(cosα)^2-(sin α)^2=2(cos α)^2-1=1-2(sin α)^2tan(2α)=2tan α/(1-tan^2 α)cot(2α)=(cot^2 α-1)/(2cot α)sec(2α)=sec^2 α/(1-tan^2 α)csc(2α)=1/2*sec ·cscα·sin(3

α)=3sin

α-4sin^3

α

=4sin

·sin(60

°+α)sin(60

°-α)cos(3

α)=4cos^3

α-3cos

α=4cos

·cos(60

°+α)cos(60

°-α)tan(3

α)=

(3tan

α-tan^3

α)/(1-3tan^2

α)=

tan

αtan(

π/3+α)tan(π/3-α)cot(3α)=(cot^3 α-3cotα)/(3cot^2 α-1)·n

倍角公式：sin(n α)=ncos^(n-1)+C(n,5)cos^(n-5) ·sin^5cos(n α)=cos^n+C(n,4)cos^(n-4) ·sin^4

·sinα-C(n,3)cos^(n-3)α-⋯α-C(n,2)cos^(n-2)α-⋯

·sin^3α·sin^2

αα·：sin(

α/2)=

±((1-cos

α)/2)cos(

α/2)=

±((1+cos

α)/2)tan(

α/2)=

±((1-cos

α)/(1+cos

α))=sin

α/(1+cos

α)=(1-cosα)/sinαcot(α/2)=±√((1+cos α)/(1-cos α))=(1+cos α)/sinα=sinα/(1-cos α)sec(α/2)=±√((2secα/(secα+1))csc(α/2)=±√((2secα/(secα-1))·辅助角公式：Asinα+Bcosα=√(A^2+B^2)sin(α+φ）（tanφ=B/A）Asinα+Bcosα=√(A^2+B^2)cos(α-φ）（tanφ=A/B）·万能公式sin(a)=(2tan(a/2))/(1+tan^2(a/2))cos(a)=(1-tan^2(a/2))/(1+tan^2(a/2))tan(a)=(2tan(a/2))/(1-tan^2(a/2))·降幂公式sin^2α=(1-cos(2α))/2=versin(2α)/2cos^2α=(1+cos(2α))/2=covers(2α)/2tan^2α=(1-cos(2α))/(1+cos(2α))·三角和的三角函数：sin(α+β+γ)=sinα·cosβ·cosγ+cosα·sinβ·cosγ+cosα·cosβ·sinγ-sinα·sinβ·sinγcos(α+β+γ)=cosα·cosβ·cosγ-cosα·sinβ·sinγ-sinα·cosβ·sinγ-sinα·sinβ·cos/r/