導(dǎo)數(shù)公式的證明(最全版)

1、導(dǎo)數(shù)的定義：f(x)=limAy/ AxAx0 (下面就不再標(biāo)明Ax0 了)用定義求導(dǎo)數(shù)公式(1) f(x)=xA n證法一：(n為自然數(shù))f(x)=lim (x+ Ax)A n-xAn/Ax=lim (x+ Ax-x)(x+ Ax)A (n-1)+x*(x+Ax)A( n-2)+xA( n-2)*(x+Ax)+xA( n-1)/Ax=lim (x+ Ax)A( n-1)+x*(x+Ax)A( n-2)+xA( n-2)*(x+Ax)+xA( n-1)=xA( n-1)+x*xA( n-2)+xA2*xA( n-3)+ xA( n-2)*x+xA( n-1)=n xA( n-1)證法二：(n為

2、任意實數(shù))f(x)=xA nInf(x)二nlnx (In f(x)=( nlnx)f(x)/f(x)二n/xf(x)二n /x*f(x)f(x)二n /x*xA nf(x)二n xA( n-1)(2) f(x)=sinxf(x)=lim (sin(x+Ax)-sinx)/ Ax=lim (sinxcos Ax+cosxsin Ax-sinx)/ Ax=lim (sinx+cosxsinAx-sinx)/ Ax=lim cosxsin Ax/ Ax=cosx(設(shè) aA Ax-1二m，貝卩 Ax=logaA(m+1)(3) f(x)二cosx f(x) =lim (cos(x+ Ax)-cosx

3、)/ Ax=lim (cosxcos Ax-sinxsin Ax-cosx)/ Ax=lim (cosx-sinxsinAx-cos)/ Ax=lim -sinxsinAx/ Ax=-s inx(4) f(x)=aAx證法一：f(x) =lim (aA(x+ Ax)-aAx)/ Ax=lim aAx*(aA Ax-1)/ Ax=lim aAx*m/logaA(m+1)=lim aAx*m/l n(m+1)/l na =lim aAx*l na*m/ln( m+1)=lim aAx* In a/(1/m)*l n( m+1)=lim aAx* lna/ln (m+1)A(1/m)=lim aAx

4、* In a/l ne=aAx*l na證法二：f(x)=aAxIn f(x)=x Inaln f(x) =xl naf (x)/f(x)=lnaf (x)=f(x)Ina f (x)=aAxIna若a=e，原函數(shù)f(x)=eAx則 f(x)二eAx*Ine=eAx(5) f(x)=logaAxf(x)=lim (IogaA(x+Ax)-IogaAx)/Ax=lim IogaA(x+Ax)/x/ Ax=lim IogaA(1+ Ax/x)/ Ax=lim In(1+ Ax/x)/(lna* Ax)=lim x*l n(1+Ax/x)/(x*l na* Ax)=lim (x/ Ax)*ln(1+

5、 Ax/x)/(x*Ina)=lim In e/(x* Ina) =1/(x*l na)=lim In(1+Ax/x)A(x/ Ax)/(x*Ina)=lim (cos(x+Ax)/s in (x+ Ax)-cosx/si nx)/ Ax若 a=e，原函數(shù) f(x)=logex=lnx則 f(x)=1/(x*lne)=1/x(6) f(x)=tanxf(x)=lim (tan(x+Ax)-tanx)/ Ax=lim (si n(x+ Ax)/cos(x+ Ax)-si nx/cosx)/ Ax=lim (sin(x+ Ax)cosx-sinxcos(x+ Ax)/( Axcosxcos(x+

6、Ax)=lim (sinxcos Axcosx+sin Axcosxcosx-sinxcosxcosAx+sinxsinxsin Ax)/( Axcosxcos(x+ Ax)=lim sin Ax/( Axcosxcos(x+ Ax)=1/(cosx)A2=secx/cosx=(secx)A2=1+(ta nx)八2(7) f(x)=cotxf(x)=lim (cot(x+Ax)-cotx)/ Ax=lim (cosx-cosxcosAx+sinxsin Ax)/( Axcosxcos(x+ Ax)=lim (cos(x+ Ax)sinx-cosxsin(x+Ax)/( Axsinxsin(x

7、+ Ax)=lim (cosxcos Axsinx-sinxsinxsinAx-cosxsinxcos Ax-cosxsin Axcosx)/( Axsinxsin(x+ Ax)=lim -sin Ax/( Axsinxsin(x+ Ax)=-1/(si nx)八2二-cscx/si nx=-(secx)八2=-1-(cotx)八2(8) f(x)=secxf(x) =lim (sec(x+ Ax)-secx)/ Ax=lim (1/cos(x+Ax)-1/cosx)/ Ax=lim (cosx-cos(x+Ax)/( Axcosxcos Ax)=lim sinxsin Ax/( Axcosx

8、cos(x+ Ax)=si nx/(cosx)八2二ta nx*secx(9) f(x)二cscx f(x) =lim (csc(x+ Ax)-cscx)/ Ax=lim (1/sin(x+Ax)-1/sinx)/ Ax=lim (sinx-sin(x+Ax)/( Axsinxsin(x+ Ax)=lim (sinx-sinxcosAx-sin Axcosx)/( Axsinxsin(x+ Ax)=lim -sin Axcosx/( Axsinxsin(x+ Ax)=-cosx/(si nx)八2二-cotx*cscx(10) f(x)=xAxIn f(x)=x Inx(In f(x)=(xl

9、 nx) f(x)/f(x)=l nx+1 f(x)=(l nx+1)*f(x) f(x)=(l nx+1)*xAx(12) h(x)=f(x)g(x) h(x)=lim (f(x+Ax)g(x+ Ax)-f(x)g(x)/ Ax=lim (f(x+Ax)-f(x)+f(x)*g(x+Ax)+(g(x+ Ax)-g(x)-g(x+Ax)*f(x)/ Ax=lim (f(x+Ax)-f(x)*g(x+ Ax)+(g(x+ Ax)-g(x)*f(x)+f(x)*g(x+Ax)-f(x)*g(x+Ax)/ Ax=lim (f(x+Ax)-f(x)*g(x+Ax)/ Ax+(g(x+ Ax)-g(x)

10、*f(x)/ Ax=f(x)g(x)+f(x)g(x)(13) h(x)=f(x)/g(x)h(x)=lim (f(x+Ax)g(x)-f(x)g(x+Ax)/( Axg(x)g(x+ Ax)=lim (f(x+Ax)/g(x+ Ax)-f(x)g(x)/ Ax=lim (f(x+Ax)-f(x)+f(x)*g(x)-(g(x+Ax)-g(x)+g(x)*f(x)/(xg(x)g(x+ )=lim (f(x+x)-f(x)*g(x)-(g(x+x)-g(x)*f(x)+f(x)g(x)-f(x)g(x)/(&g(x)g(x+ &)=lim (f(x+ Zx)-f(x)*g(x)/(xg(x)g

11、(x+ Zx)-(g(x+ x)-g(x)*f(x)/(xg(x)g(x+ &)=f(x)g(x)/(g(x)*g(x)-f(x)g(x)/(g(x)*g(x)=f(x)g(x)-f(x)g(x)/(g(x)*g(x)x(14) h(x)=f(g(x) h(x)=lim f(g(x+x)-f(g(x)/ =lim f(g(x+x)-g(x)+g(x)-f(g(x)/x(另 g(x)=u , g(x+ Zx)-g(x)= u)=lim (f(u+u)-f(u)/ Zx=lim (f(u+u)-f(u)* u/( Zx* u)=lim f(u)*Au/ Ax=lim f(u)*(g(x+Ax)-g

12、(x)/ Ax二f(u)*g(x)=f(g(x)g(x)(反三角函數(shù)的導(dǎo)數(shù)與三角函數(shù)的導(dǎo)數(shù)的乘積為1，因為函數(shù)與反函數(shù)關(guān)于y=x對稱，所以導(dǎo)數(shù)也關(guān)于y=x對稱，所以導(dǎo)數(shù)的乘積為1)(15) y=f(x)二arcsinx貝卩siny=x(siny )=cosy所以(arcsi nx)=1/(si ny)=1/cosy=1/ v1-(siny)八2(siny=x)=1/ v1-xA2即 f(x)=1/ V1-xA2(16) y=f(x)=arcta nx貝卩tany=x(tan y)=1+(ta nyF2=1+xA2所以(arcta nx)=1/1+xT即 f(x)= 1/1+xA2總結(jié)一下(xAn )二nxA(n-1)(sinx )二cosx(cosx ) =-sinx(aAx ) =aAxIna(eAx ) =eAx(logaAx ) =1/(x1 na)(I nx ) =1/x(ta nx)=(secx)A2=1+(ta nx)A2(c