三角函数导数微分积分

三角函数 诱导公式tgA=tanA=sinacossin(sin(a)sinasin(-a)=2cosasin(+a)=2cosasin(π-a)=sin(π+a)=sina-sinacos(-a)=cosa cos(-a)=2sinacos(+a)=2-sinacos(π-a)=cos(π+a)=-cosa-cosasin(A+B)=sinAcosB+cosAsinBcos(A+B)=cosAcosB-sinAsinBsin(A+B)=sinAcosB+cosAsinBcos(A+B)=cosAcosB-sinAsinBsin(A-B)=sinAcosB-cosAsinBtan(A+B)=tanAtanB1-tanAtanBcos(A-B)=cosAcosB+sinAsinBcot(A+B)=cotAcotB-1cotBcotAtan(A-B)=tanAtanB1tanAtanBcot(A-B)=cotAcotB1cotBcotA倍角公式三倍角公式半角公式tan2A=2tanA1tansin3A=3sinA-4(sinA)3sin( )=2A1cos2Sin2A=2SinA•CosAcos3A=4(cosA)3-3cosAcos(A)=21cos2Cos2A=tan3a=CosA-SinA=2CosA-1=1-2sinAtan(A)=22222tana·tan(+a)·tan(-a)1cosA1cosA33cot(A)=21cosA1cosAtan(A)=1cosA=2sinAsinA1和差化积sina+sinb=2sinabcosab2 2

积化和差sinasinb=-1[cos(a+b)-cos(a-b)]2sina-sinb=2cosabsinab2 2cosa+cosb=2cosabcosab2 2cosa-cosb=-2sinabsinab2 2

cosacosbsinacosbcosasinb

1[cos(a+b)+cos(a-b)]21[sin(a+b)+sin(a-b)]21[sin(a+b)-sin(a-b)]2tana+tanb=

sin(ab)cosacos万能公式sina=sina=2tana2a2cosa=1(tana2a2)2tana=2tan1(tan)21(tan)21(tana2a2)2其他非重点三角函数csc(a)=csc(a)=1sinsec(a)=1coscot2a1csc2atan2a1sec2a1cos2a双曲函数sinh(a)=sinh(a)=ea-e-a2cosh(a)=eae-a2tgh(a)=sinh(a)cosh(a)n2n2sinx~xarcsinx~x~arctanxx)~xex1~xtanx~xx21cosx~x)u1~uxn1x1~xax1~xlnasinx 1

两个重要的极限lim x 1x)xex0 x导数、微分、积分函数的和差积商求导函数的和差积商求导法则uv'u'v'函数的和差积商微分法则d(uv)dudv函数的和差积商求导法则xudxxu1u1c(u1)Cu'Cu'uv'u'vuv'd(Cu)Cdud(uv)vduudvu'vu'vuv'v2d( )vuvduv2kfkffff'fux高阶导数yf(xy

fxydydx

y

fx的导数称为 d2

ddy二阶导数，记作y"

y''或 ＝ 类似的二阶导数的导数称为三阶导数三dx2 dxdxn阶导数记作dny导数公式C导数公式C'0(xu)'uxu1微分公式dyf'(x)dx积分公式kdxkxCd(xu)uxudxxdxuxu1u1C( )'x11x2d()dx1dxlnxCx(sinx)'cosxd(sinx)cosxdxcosxdxsinxC(cosx)'sinxd(cosx)sinxdxsinxdxcosxC(tanx)'sec2xd(tanx)sec2xdx1cos2xdxsec2xdxtanxCtanxdxlncosxC(cotx)'csc2xd(cotx)csc2xdx1sin2xdxcsc2xdxcotxCcotxdxlnsinxC(secx)'secxtanx d(secx)secxtanxdx secxdxlnsecxtanxCsecxtanxdxsecxC(cscx)'cscxcotx d(cscx)cscxcotxdx cscxdxlncscxcotxCcscxcotxdxcscxC(ax)'axlna d(ax)axlnadx axaxdx

lnaC(ex)'ex d(ex)exdx exdxexC(loga

x)'1

1xlna

d(loga

x)1

1 xlna

dxC1(lnx)'x1

d(lnx)xdx1

xdxlnxC1(arcsinx)'

1x2

d(arcsinx)

1x

dx

1x

dxarcsinxC(arccosx)'

11x21

d(arccosx)

1 dx1x21

dxC 1 dxarctanxC(arctanx)'

1x2

d(arctanx) dx1x2

1x2(arccotx)'(shx)'chx(chx)'shx

11x2

d(arccot)d()dx

1 dx1x2

dxCshxdxchxCchxdxshxC1

dx

1arctanxC(thx)'

ch2x

a2x2 a a(arshx)'

11x2

1 dx