導(dǎo)數(shù)公式的證明(最全版)

1、僅供個人參考導(dǎo)數(shù)的定義：f(x)=lim勺/ AxAxT (下面就不再標(biāo)明A x 0 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))For pers onal use only in study and research; not for commercialusef(x)=lim (x+ Ax)An-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)/ AxFor pers onal use only in study and research; not for co

2、mmercialuse=lim (x+ Ax)A( n-1)+x*(x+ Ax)A( n-2)+xA( n-2)*(x+ Ax)+xA( n-1)=xA( n-1)+x*x( n-2)+x2*xA( n-3)+ .xA( n-2)*x+x( n-1)=n xA( n-1)commercialFor personal use only in study and research; not foruse證法二：( n 為任意實數(shù))f(x)=xAnlnf(x)=nlnx(lnf(x)=(nlnx)f(x)/f(x)=n/xf(x)=n/x*f(x)f(x)=n/x*xAnf(x)=nxA(n-1)(

3、2) f(x)=sinxf(x)=lim (sin(x+ Ax)-sinx)/ Ax=lim (sinxcos Ax+cosxsin Ax-sinx)/ Ax=lim (sinx+cosxsin Ax-sinx)/Ax=lim cosxsin Ax/Ax=cosx(3) f(x)=cosxf(x)=lim (cos(x+ Ax)-cosx)/ Ax=lim (cosxcos Ax-sinxsin Ax-cosx)/Ax=lim (cosx-sinxsin Ax-cos)/ Ax=lim -sinxsin Ax/Ax=-sinx(4) f(x)=aAx證法一：f(x)=lim (aA(x+ Ax

4、)-aAx)/ Ax=lim aAx*(aA Ax-1)/ Ax(設(shè) aA Ax-仁m，則 Ax=logaA(m+1)=lim aAx*m/logaA(m+1)=lim aAx*m/ln(m+1)/lna=lim aAx*lna*m/ln(m+1)=lim aAx*lna/(1/m)*ln(m+1)=lim aAx*lna/ln(m+1)A(1/m)=lim aAx*lna/lne=aAx*lna證法二：f(x)=aAxlnf(x)=xlnalnf(x) =xlna f (x)/f(x)=lnaf (x)=f(x)lnaf (x)=aAxlna若 a=e ，原函數(shù) f(x)=eAx則 f(x)

5、=eAx*lne=eAx(5) f(x)=logaAxf(x)=lim (logaA(x+ Ax)-logaAx)/ Ax=lim logaA(x+ Ax)/x/ Ax=lim logaA(1+ Ax/x)/ Ax=lim In(1+ Ax/x)/(lna* Ax)=lim x*ln(1+ Ax/x)/(x*Ina* Ax)=lim (x/Ax)*ln(1+ Ax/x)/(x*lna)=lim In(1+ Ax/x)A(x/ Ax)/(x*Ina)=lim lne/(x*lna)=1/(x*lna)若 a = e ，原函數(shù) f(x)=logeAx=lnx則 f(x)=1/(x*lne)=1/x

6、(6) f(x)=tanxf(x) =lim (tan(x+ Ax)-tanx)/ Ax=lim (sin(x+ Ax)/cos(x+ Ax)-sinx/cosx)/ Ax=lim (sin(x+ Ax)cosx-sinxcos(x+ Ax)/(Axcosxcos(x+ Ax)=lim (sinxcos Axcosx+sin Axcosxcosx-sinxcosxcos Ax+sinxsinxsin x)/( Axcosxcos(x+ Ax)=lim sin x/( xcosxcos(x+ x)=1/(cosx)A2=secx/cosx=(secx)A2=1+(ta nx)八2(7) f(x)

7、=cotxf(x)=lim (cot(x+ Ax)-cotx)/Ax=lim (cos(x+ Ax)/sin(x+ Ax)-cosx/sinx)/ Ax=lim (cos(x+ Ax)sinx-cosxsin(x+ Ax)/(Axsinxsin(x+ Ax)=lim (cosxcos Axsinx-sinxsinxsin Ax-cosxsinxcos Ax-cosxsin Axcosx)/( Axsinxsin(x+ Ax)=lim -sin Ax/( Axsinxsin(x+ Ax)=-1/(sinx)A2=-cscx/sinx=-(secx)A2=-1-(cotx)A2(8) f(x)=s

8、ecxf(x)=lim (sec(x+ Ax)-secx)/ Ax=lim (1/cos(x+ Ax)-1/cosx)/ Ax=lim (cosx-cos(x+ Ax)/( Axcosxcos Ax)=lim (cosx-cosxcos Ax+sinxsin Ax)/( Axcosxcos(x+ Ax)=lim sinxsin Ax/( Axcosxcos(x+ Ax)=si nx/(cosx)八2二ta nx*secx(9) f(x)=cscxf(x)=lim (csc(x+ Ax)-cscx)/Ax=lim (1/sin(x+ Ax)-1/sinx)/ Ax=lim (sinx-sin(x

9、+ Ax)/(Axsinxsin(x+ Ax)=lim (sinx-sinxcos Ax-sin Axcosx)/( Axsinxsin(x+ Ax)=lim -sin Axcosx/(Axsinxsin(x+ Ax)二-cosx/(si nx)八2二-cotx*cscx(10) f(x)=xAx lnf(x)=xlnx (lnf(x)=(xlnx) f(x)/f(x)=lnx+1 f(x)=(lnx+1)*f(x) f(x)=(lnx+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)-

10、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)*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+ Ax)-f(x)g(x)/ Ax=lim (f(x+ Ax)g(x)-f(x)g(x+ Ax)/(Axg(

11、x)g(x+ Ax)=lim (f(x+ Ax)-f(x)+f(x)*g(x)-(g(x+ Ax)-g(x)+g(x)*f(x)/( Axg(x)g(x+ A x)=lim (f(x+ Ax)-f(x)*g(x)-(g(x+ Ax)-g(x)*f(x)+f(x)g(x)-f(x)g(x)/(Axg(x)g(x+ Ax)=lim (f(x+ Ax)-f(x)*g(x)/( Axg(x)g(x+ Ax)-(g(x+ Ax)-g(x)*f(x)/( A xg(x)g(x+ Ax)=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

12、(x)*g(x)x(14) h(x)=f(g(x)h(x)=lim f(g(x+ Ax)-f(g(x)/ Ax=lim f(g(x+ Ax)-g(x)+g(x)-f(g(x)/ Ax(另 g(x)=u，g(x+Ax)-g(x)= Au)=lim (f(u+ Au)-f(u)/Ax=lim (f(u+ Au)-f(u)* Au/( Ax* Au)=lim f(u)* Au/Ax=lim f(u)*(g(x+ Ax)-g(x)/ Ax=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

13、積為 1)(15) y=f(x)=arcsinx則 siny=x(siny)=cosy所以(arcsinx)=1/(siny)=1/cosy=1/-(si ny )八2(siny=x)=1/VI-xA2即 f(x)=1/ V-xA2(16) y=f(x)=arctanx 則 tany=x (tany)=1+(tany)A2=1+xA2所以(arctanx)=1/1+xA2 即 f(x)= 1/1+xA2總結(jié)一下（xAn ）=nxA（n-1）（sinx）=cosx（cosx）=-sinx(aAx) =aAxIna(eAx) =eAx( IogaAx )=1/(xIna)( Inx)=1/x(ta

14、nx)=(secx)A2=1+(tanx)A2(cotx)=-(cscx)A2=-1-(cotx)A2(secx)=tanx*secx(cscx)=-cotx*cscx(xAx)=(Inx+1)*xAx(arcsinx)=1/ V-X2(arctanx)=1/1+xA2f(x)g(x)=f(x)g(x)+f(x)g(x)f(x)/g(x)=f(x)g(x)-f(x)g(x)/(g(x)*g(x)f(g(x)=f(g(x)g(x)僅供個人用于學(xué)習(xí)、研究；不得用于商業(yè)用途For personal use only in study and research; not for commercial use.Nur f u r den pers?nlichen f u r Studien, Forschung, zu kommerziellen Zwecke