导数公式的证明最全版.docx
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导数公式的证明最全版
导数的定义:
f'(*)=limΔy/Δ*
Δ*→0〔下面就不再标明Δ*→0了〕
用定义求导数公式
〔1〕f(*)=*^n
证法一:
〔n为自然数〕
f'(*)
=lim[(*+Δ*)^n-*^n]/Δ*
=lim(*+Δ*-*)[(*+Δ*)^(n-1)+**(*+Δ*)^(n-2)+...+*^(n-2)*(*+Δ*)+*^(n-1)]/Δ*
=lim[(*+Δ*)^(n-1)+**(*+Δ*)^(n-2)+...+*^(n-2)*(*+Δ*)+*^(n-1)]
=*^(n-1)+***^(n-2)+*^2**^(n-3)+...*^(n-2)**+*^(n-1)
=n*^(n-1)
证法二:
〔n为任意实数〕
f(*)=*^n
lnf(*)=nln*
(lnf(*))'=(nln*)'
f'(*)/f(*)=n/*
f'(*)=n/**f(*)
f'(*)=n/***^n
f'(*)=n*^(n-1)
〔2〕f(*)=sin*
f'(*)
=lim(sin(*+Δ*)-sin*)/Δ*
=lim(sin*cosΔ*+cos*sinΔ*-sin*)/Δ*
=lim(sin*+cos*sinΔ*-sin*)/Δ*
=limcos*sinΔ*/Δ*
=cos*
〔3〕f(*)=cos*
f'(*)
=lim(cos(*+Δ*)-cos*)/Δ*
=lim(cos*cosΔ*-sin*sinΔ*-cos*)/Δ*
=lim(cos*-sin*sinΔ*-cos)/Δ*
=lim-sin*sinΔ*/Δ*
=-sin*
〔4〕f(*)=a^*
证法一:
f'(*)
=lim(a^(*+Δ*)-a^*)/Δ*
=lima^**(a^Δ*-1)/Δ*
〔设a^Δ*-1=m,则Δ*=loga^(m+1)〕
=lima^**m/loga^(m+1)
=lima^**m/[ln(m+1)/lna]
=lima^**lna*m/ln(m+1)
=lima^**lna/[(1/m)*ln(m+1)]
=lima^**lna/ln[(m+1)^(1/m)]
=lima^**lna/lne
=a^**lna
证法二:
f(*)=a^*
lnf(*)=*lna
[lnf(*)]'=[*lna]'
f'(*)/f(*)=lna
f'(*)=f(*)lna
f'(*)=a^*lna
假设a=e,原函数f(*)=e^*
则f'(*)=e^**lne=e^*
〔5〕f(*)=loga^*
f'(*)
=lim(loga^(*+Δ*)-loga^*)/Δ*
=limloga^[(*+Δ*)/*]/Δ*
=limloga^(1+Δ*/*)/Δ*
=limln(1+Δ*/*)/(lna*Δ*)
=lim**ln(1+Δ*/*)/(**lna*Δ*)
=lim(*/Δ*)*ln(1+Δ*/*)/(**lna)
=limln[(1+Δ*/*)^(*/Δ*)]/(**lna)
=limlne/(**lna)
=1/(**lna)
假设a=e,原函数f(*)=loge^*=ln*
则f'(*)=1/(**lne)=1/*
〔6〕f(*)=tan*
f'(*)
=lim(tan(*+Δ*)-tan*)/Δ*
=lim(sin(*+Δ*)/cos(*+Δ*)-sin*/cos*)/Δ*
=lim(sin(*+Δ*)cos*-sin*cos(*+Δ*)/(Δ*cos*cos(*+Δ*))
=lim(sin*cosΔ*cos*+sinΔ*cos*cos*-sin*cos*cosΔ*+sin*sin*sinΔ*)/(Δ*cos*cos(*+Δ*))
=limsinΔ*/(Δ*cos*cos(*+Δ*))
=1/(cos*)^2=sec*/cos*=(sec*)^2=1+(tan*)^2
〔7〕f(*)=cot*
f'(*)
=lim(cot(*+Δ*)-cot*)/Δ*
=lim(cos(*+Δ*)/sin(*+Δ*)-cos*/sin*)/Δ*
=lim(cos(*+Δ*)sin*-cos*sin(*+Δ*))/(Δ*sin*sin(*+Δ*))
=lim(cos*cosΔ*sin*-sin*sin*sinΔ*-cos*sin*cosΔ*-cos*sinΔ*cos*)/(Δ*sin*sin(*+Δ*))
=lim-sinΔ*/(Δ*sin*sin(*+Δ*))
=-1/(sin*)^2=-csc*/sin*=-(sec*)^2=-1-(cot*)^2
〔8〕f(*)=sec*
f'(*)
=lim(sec(*+Δ*)-sec*)/Δ*
=lim(1/cos(*+Δ*)-1/cos*)/Δ*
=lim(cos*-cos(*+Δ*)/(Δ*cos*cosΔ*)
=lim(cos*-cos*cosΔ*+sin*sinΔ*)/(Δ*cos*cos(*+Δ*))
=limsin*sinΔ*/(Δ*cos*cos(*+Δ*))
=sin*/(cos*)^2=tan**sec*
〔9〕f(*)=csc*
f'(*)
=lim(csc(*+Δ*)-csc*)/Δ*
=lim(1/sin(*+Δ*)-1/sin*)/Δ*
=lim(sin*-sin(*+Δ*))/(Δ*sin*sin(*+Δ*))
=lim(sin*-sin*cosΔ*-sinΔ*cos*)/(Δ*sin*sin(*+Δ*))
=lim-sinΔ*cos*/(Δ*sin*sin(*+Δ*))
=-cos*/(sin*)^2=-cot**csc*
〔10〕f(*)=*^*
lnf(*)=*ln*
(lnf(*))'=(*ln*)'
f'(*)/f(*)=ln*+1
f'(*)=(ln*+1)*f(*)
f'(*)=(ln*+1)**^*
〔12〕h(*)=f(*)g(*)
h'(*)
=lim(f(*+Δ*)g(*+Δ*)-f(*)g(*))/Δ*
=lim[(f(*+Δ*)-f(*)+f(*))*g(*+Δ*)+(g(*+Δ*)-g(*)-g(*+Δ*))*f(*)]/Δ*
=lim[(f(*+Δ*)-f(*))*g(*+Δ*)+(g(*+Δ*)-g(*))*f(*)+f(*)*g(*+Δ*)-f(*)*g(*+Δ*)]/Δ*
=lim(f(*+Δ*)-f(*))*g(*+Δ*)/Δ*+(g(*+Δ*)-g(*))*f(*)/Δ*
=f'(*)g(*)+f(*)g'(*)
〔13〕h(*)=f(*)/g(*)
h'(*)
=lim(f(*+Δ*)/g(*+Δ*)-f(*)g(*))/Δ*
=lim(f(*+Δ*)g(*)-f(*)g(*+Δ*))/(Δ*g(*)g(*+Δ*))
=lim[(f(*+Δ*)-f(*)+f(*))*g(*)-(g(*+Δ*)-g(*)+g(*))*f(*)]/(Δ*g(*)g(*+Δ*))
=lim[(f(*+Δ*)-f(*))*g(*)-(g(*+Δ*)-g(*))*f(*)+f(*)g(*)-f(*)g(*)]/(Δ*g(*)g(*+Δ*))
=lim(f(*+Δ*)-f(*))*g(*)/(Δ*g(*)g(*+Δ*))-(g(*+Δ*)-g(*))*f(*)/(Δ*g(*)g(*+Δ*))
=f'(*)g(*)/(g(*)*g(*))-f(*)g'(*)/(g(*)*g(*))
=[f'(*)g(*)-f(*)g'(*)]/(g(*)*g(*))*
〔14〕h(*)=f(g(*))
h'(*)
=lim[f(g(*+Δ*))-f(g(*))]/Δ*
=lim[f(g(*+Δ*)-g(*)+g(*))-f(g(*))]/Δ*
〔另g(*)=u,g(*+Δ*)-g(*)=Δu〕
=lim(f(u+Δu)-f(u))/Δ*
=lim(f(u+Δu)-f(u))*Δu/(Δ**Δu)
=limf'(u)*Δu/Δ*
=limf'(u)*(g(*+Δ*)-g(*))/Δ*
=f'(u)*g'(*)=f'(g(*))g'(*)
(反三角函数的导数与三角函数的导数的乘积为1,因为函数与反函数关于y=*对称,所以导数也关于y=*对称,所以导数的乘积为1)
〔15〕y=f(*)=arcsin*
则siny=*
(siny)'=cosy
所以
(arcsin*)'=1/(siny)'=1/cosy
=1/√1-(siny)^2
(siny=*)
=1/√1-*^2
即f'(*)=1/√1-*^2
(16)y=f(*)=arctan*
则tany=*
(tany)'=1+(tany)^2=1+*^2
所以
(arctan*)'=1/1+*^2
即f'(*)=1/1+*^2
总结一下
〔*^n〕'=n*^(n-1)
〔sin*〕'=cos*
〔cos*〕'=-sin*
〔a^*〕'=a^*lna
〔e^*〕'=e^*
〔loga^*〕'=1/(*lna)
〔ln*〕'=1/*
(tan*)'=(sec*)^2=1+(tan*)^2
(cot*)'=-(csc*)^2=-1-(cot*)^2
(sec*)'=tan**sec*
(csc*)'=-cot**csc*
(*^*)'=(ln*+1)**^*
(arcsin*)'=1/√1-*^2
(arctan*)'=1/1+*^2
[f(*)g(*)]'=f'(*)g(*)+f(*)g'(*)
[f(*)/g(*)]'=[f'(*)g(*)-f(*)g'(*)]/(g(*)*g(*))
[f(g(*))]'=f'(g(*))g'(*)