Question

高中常用函数的导函数。

y=sin x,y=cosx.

Answer 1

函数导数公式
列举几基本函数导数及推导程:
1.y=c(c数)
y'=0
2.y=x^n
y'=nx^(n-1)
3.y=a^x
y'=a^xlna
y=e^x
y'=e^x
4.y=logax
y'=logae/x
y=lnx
y'=1/x
5.y=sinx
y'=cosx
6.y=cosx
y'=-sinx
7.y=tanx
y'=1/cos^2x
8.y=cotx
y'=-1/sin^2x
9.y=arcsinx
y'=1/√1-x^2
10.y=arccosx
y'=-1/√1-x^2
11.y=arctanx
y'=1/1+x^2
12.y=arccotx
y'=-1/1+x^2
推导程几见公式需要用:
1.y=f[g(x)],y'=f'[g(x)]&8226;g'(x)『f'[g(x)]g(x)看作整变量g'(x)x看作变量』
2.y=u/v,y'=(u'v-uv')/v^2
3.y=f(x)反函数x=g(y)则y'=1/x'
证:1.显易见y=c条平行于x轴直线所处处切线都平行于x故斜率0用导数定义做:y=c,⊿y=c-c=0,lim⊿x→0⊿y/⊿x=0
2.推导暂且证根据导数定义推导能推广n任意实数般情况
y=e^x
y'=e^xy=lnx
y'=1/x两结能用复合函数求导给予证明
3.y=a^x,
⊿y=a^(x+⊿x)-a^x=a^x(a^⊿x-1)
⊿y/⊿x=a^x(a^⊿x-1)/⊿x
直接令⊿x→0能导导函数必须设辅助函数β=a^⊿x-1通换元进行计算由设辅助函数知道:⊿x=loga(1+β)
所(a^⊿x-1)/⊿x=β/loga(1+β)=1/loga(1+β)^1/β
显⊿x→0β趋向于0limβ→0(1+β)^1/β=e,所limβ→01/loga(1+β)^1/β=1/logae=lna
结代入lim⊿x→0⊿y/⊿x=lim⊿x→0a^x(a^⊿x-1)/⊿xlim⊿x→0⊿y/⊿x=a^xlna
知道a=ey=e^x
y'=e^x
4.y=logax
⊿y=loga(x+⊿x)-logax=loga(x+⊿x)/x=loga[(1+⊿x/x)^x]/x
⊿y/⊿x=loga[(1+⊿x/x)^(x/⊿x)]/x
⊿x→0⊿x/x趋向于0x/⊿x趋向于∞所lim⊿x→0loga(1+⊿x/x)^(x/⊿x)=logae,所
lim⊿x→0⊿y/⊿x=logae/x
知道a=ey=lnx
y'=1/x
进行y=x^n
y'=nx^(n-1)推导y=x^n,所y=e^ln(x^n)=e^nlnx,
所y'=e^nlnx&8226;(nlnx)'=x^n&8226;n/x=nx^(n-1)
5.y=sinx
⊿y=sin(x+⊿x)-sinx=2cos(x+⊿x/2)sin(⊿x/2)
⊿y/⊿x=2cos(x+⊿x/2)sin(⊿x/2)/⊿x=cos(x+⊿x/2)sin(⊿x/2)/(⊿x/2)
所lim⊿x→0⊿y/⊿x=lim⊿x→0cos(x+⊿x/2)&8226;lim⊿x→0sin(⊿x/2)/(⊿x/2)=cosx
6.类似导y=cosx
y'=-sinx
7.y=tanx=sinx/cosx
y'=[(sinx)'cosx-sinx(cos)']/cos^2x=(cos^2x+sin^2x)/cos^2x=1/cos^2x
8.y=cotx=cosx/sinx
y'=[(cosx)'sinx-cosx(sinx)']/sin^2x=-1/sin^2x
9.y=arcsinx
x=siny
x'=cosy
y'=1/x'=1/cosy=1/√1-sin^2y=1/√1-x^2
10.y=arccosx
x=cosy
x'=-siny
y'=1/x'=-1/siny=-1/√1-cos^2y=-1/√1-x^2
11.y=arctanx
x=tany
x'=1/cos^2y
y'=1/x'=cos^2y=1/sec^2y=1/1+tan^2x=1/1+x^2
12.y=arccotx
x=coty
x'=-1/sin^2y
y'=1/x'=-sin^2y=-1/csc^2y=-1/1+cot^2y=-1/1+x^2
另外双曲函数shx,chx,thx等及反双曲函数arshx,archx,arthx等其较复杂复合函数求导通查阅导数表运用公式与
4.y=u土v,y'=u'土v'
5.y=uv,y=u'v+uv'