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导数的定义:f'(x)=lim ΔyΔx
Δx→0(下面就不再标明Δx→0了)
用定义求导数公式
(1)f(x)=x^n
证法一:(n为自然数)
For personal use only in study and research; not for commercial
use
f'(x)
=lim [(x+Δx)^n-x^n]Δx
=lim (x+Δx -x)[(x+Δx)^(n-1)+x*(x+Δx)^(n-2)+...+x^(n-2)*(x+Δx)+x^(n-1)]Δx
For personal use only in study and research; not for commercial
use
=lim [(x+Δx)^(n-1)+x*(x+Δx)^(n-2)+...+x^(n-2)* (x+Δx)+x^(n-1)]
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=x^(n-1)+x*x^(n-2)+x^2*x^(n-3)+ ...x^(n-2)*x+x^(n-1)
=nx^(n-1)
For personal use only in study and research; not for commercial
use
证法二:(n为任意实数)
f(x)=x^n
lnf(x)=nlnx
(lnf(x))'=(nlnx)'
f'(x)f(x)=nx
f'(x)=nx*f(x)
f'(x)=nx*x^n
f'(x)=nx^(n-1)
(2)f(x)=sinx
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f'(x)
=lim (sin(x+Δx)-sinx)Δx
=lim (sinxcosΔx+cosxsinΔx- sinx)Δx
=lim (sinx+cosxsinΔx-sinx)Δx
=lim cosxsinΔxΔx
=cosx
(3)f(x)=cosx
f'(x)
=lim (cos(x+Δx)-cosx)Δx
=lim (cosxcosΔx-sinxsinΔx-cosx)Δx
=lim (cosx- sinxsinΔx-cos)Δx
=lim -sinxsinΔxΔx
=-sinx
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(4)f(x)=a^x
证法一:
f'(x)
=lim (a^(x+Δx)-a^x)Δx
=lim a^x*(a^Δx-1)Δx
(设a^Δx-1=m,则Δx=loga^(m+1))
=lim a^x*mloga^(m+1)
=lim a^x*m[ln(m+1)lna]
=lim a^x*lna*mln(m+1)
=lim a^x*lna[(1m)*ln(m+1)]
=lim a^x*lnaln[(m+1)^(1m)]
=lim a^x*lnalne
=a^x*lna
证法二:
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f(x)=a^x
lnf(x)=xlna
[lnf(x)] '=[xlna] '
f' (x)f(x)=lna
f' (x)=f(x)lna
f' (x)=a^xlna
若a=e,原函数f(x)=e^x
则f'(x)=e^x*lne=e^x
(5)f(x)=loga^x
f'(x)
=lim (loga^(x+Δx)-loga^x)Δx
=lim loga^[(x+Δx)x]Δx
=lim loga^(1+Δxx)Δx
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=lim ln(1+Δxx)(lna*Δx)
=lim x*ln(1+Δxx)(x*lna*Δx)
=lim (xΔx)*ln(1+Δxx)(x*lna)
=lim ln[(1+Δxx)^(xΔx)](x*lna)
=lim lne(x*lna)
=1(x*lna)
若a=e,原函数f(x)=loge^x=lnx
则f'(x)=1(x*lne)=1x
(6)f(x)=tanx
f'(x)
=lim (tan(x+Δx)-tanx)Δx
=lim (sin(x+Δx)cos(x+Δx)-sinxcosx)Δx
=lim (sin(x+Δx)cosx-sinxcos(x+Δx)(Δxcosxcos(x+Δx))
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=lim (sinxcosΔxcosx+sinΔxcosxcosx- sinxcosxcosΔx+sinxsinxsinΔ
x)(Δxcosxcos(x+Δx))
=lim sinΔx(Δxcosxcos(x+Δx))
=1(cosx)^2=secxcosx=(secx)^2=1+(tanx)^2
(7)f(x)=cotx
f'(x)
=lim (cot(x+Δx)-cotx)Δx
=lim (cos(x+Δx)sin(x+Δx)-cosxsinx)Δx
=lim (cos(x+Δx)sinx-cosxsin(x+Δx))(Δxsinxsin(x+Δx))
=lim (cosxcosΔxsinx-sinxsinxsinΔx- cosxsinxcosΔx-cosxsinΔ
xcosx)(Δxsinxsin(x+Δx))
=lim -sinΔx(Δxsinxsin(x+Δx))
=-1(sinx)^2=-cscxsinx=-(secx)^2=-1-(cotx)^2
(8)f(x)=secx
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f'(x)
=lim (sec(x+Δx)-secx)Δx
=lim (1cos(x+Δx)-1cosx)Δx
=lim (cosx- cos(x+Δx)(ΔxcosxcosΔx)
=lim (cosx- cosxcosΔx+sinxsinΔx)(Δxcosxcos(x+Δx))
=lim sinxsinΔx(Δxcosxcos(x+Δx))
=sinx(cosx)^2=tanx*secx
(9)f(x)=cscx
f'(x)
=lim (csc(x+Δx)-cscx)Δx
=lim (1sin(x+Δx)-1sinx)Δx
=lim (sinx- sin(x+Δx))(Δxsinxsin(x+Δx))
=lim (sinx- sinxcosΔx-sinΔxcosx)(Δxsinxsin(x+Δx))
=lim -sinΔxcosx(Δxsinxsin(x+Δx))
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