导数公式地证明(

1、实用标准文案导数的定义：f(x)=limAy/ AxAxt0 (下面就不再标明A xf 0 了)用定义求导数公式(1) f(x)=xA n证法一：(n为自然数)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

2、)证法二：(n为任意实数)f(x)=xA nlnf(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(3) f(x)二cosxf(x)=lim (cos(x+ Ax)-cosx)/ Ax=lim (co

3、sxcos Ax-sinxsin Ax-cosx)/ Ax=lim (cosx-sinxsinAx-cos)/ Ax=lim -sinxsinAx/ Ax=-s inx(4) f(x)=aAx证法一：f(x)=lim (aA(x+Ax)-aAx)/ Ax=lim aAx*(aAAx-1)/ Ax(设 aA Ax-1二m，则 Ax=logaA(m+1)=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

4、/m)=lim aAx* In a/l ne=aAx*l na证法二：f(x)=aAxIn f(x)=x Inaln f(x) '=xl na' f (x)/f(x)=lna f (x)=f(x)Ina f (x)=aAxIna若a=e，原函数f(x)=eAx 则 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 n

5、a*Ax)=lim (x/ Ax)*ln(1+ Ax/x)/(x*Ina)=lim In(1+Ax/x)A(x/ Ax)/(x*Ina)=lim In e/(x* Ina)=1/(x* In a)若 a=e，原函数 f(x)=logex=lnx则 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+ Ax)=lim (sinxcos Axco

6、sx+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 (cos(x+Ax)/s in (x+ Ax)-cosx/si nx)/ Ax=lim (cos(x+Ax)sinx-cosxsin(x+Ax)/( Axsinxsin(x+ Ax)=lim (cosxcos Axsinx-sin

7、xsinxsinAx-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 (cosx-cosxcosAx+sinxsin Ax)/( Axcosxcos(x+ Ax)=l

8、im sinxsin Ax/( Axcosxcos(x+ Ax)=si nx/(cosx)八2二ta nx*secx(9) f(x)二cscxf(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

9、)=x Inx(In f(x)'=(xl 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+A

10、x)-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(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

11、(x)/(&g(x)g(x+ &)=lim (f(x+ Zx)-f(x)*g(x)/(xg(x)g(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 (

12、f(u+u)-f(u)/ =lim (f(u+u)-f(u)* u/( Zx* u)=lim f(u)*Au/ Ax=lim f(u)*(g(x+Ax)-g(x)/ Ax二f(u)*g'(x)=f(g(x)g'(x)(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=x对称，所以导数也关于y=x对称，所以导数的乘积为1)(15) y=f(x)二arcsinx贝卩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)

13、=1/ V1-xA2(16) y=f(x)=arcta nx贝卩tany=x(tan y)'=1+(ta nyF2=1+xA2所以(arcta nx)'=1/1+xT即 f(x)= 1/1+xA2总结一下(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(cotx)'=-(cscx)A2=-1-(cotx)A2(secx)'=ta nx*secx(c