导数公式的证明最全版.docx

1、导数公式的证明最全版导数的定义：f(x)=lim 刖 bx x宀0(下面就不再标明 x t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+ -xx)An/ x=lim (x+ -x)x(x+ x)-A1()n+x*(x+ x)A(n-2)+.+xA(n-2)*(x+ x)+x-A1(n)/ x=lim (x+ x)A-(1n)+x*(x+ x)-A2(n)+.+xA(n-2)*(x+ x)+x-A1(n) =xA(n-1)+x*xA(n-2)+xA2*xA(n-3)+ .xA(n-2)*x+xA(n-1)=nxA(n-1) 证法二：( n 为任意

2、实数)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)二n xA( n-1)(2) f(x)=sinxf(x)=lim (si n(x+ -six)x)/ Ax=lim (sinxcos A x+cosxsinx)/AxX x=lim (sinx+cosxsin -sinx)/x Ax=lim cosxsin A x/ Ax=cosx( 3) f(x)=cosxf(x)=lim (cos(x+ -ctos)/ Ax=lim (cosxcos -sinxsin -Acxx)/ Ax=lim (c

3、osx-sinxsin -Aos)/ Ax=lim -sinxsin A x/ Ax=-sinx( 4) f(x)=aAx证法一：f(x)=lim (aA(x+ -aAx)/ Ax=lim aAx*(aA -1)/x Ax(设 aAA-1= m,贝卩 Ax=logaA(m+=lim aAx*m/logaA(m+1)=lim aAx*m/l n(m+1)/l na=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证法二：f(x)=aAxlnf(x)=x

4、lnalnf(x) =xlna f (x)/f(x)=lnaf (x)=f(x)lnaf (x)=aAxlna若a=e,原函数f(x)=eAx则 f(x)=eAx*lne=eAx( 5)f(x)=logaAxf(x)=lim (logaA(x+ -logxAx)/ Ax=lim logaA(x+ A x)/x/ Ax=lim logaA(1+ A x/x)/ Axlim In (1+ x/x)/(l na* x)lim x*l n(1+ x/x)/(x*l na* x)lim (x/ x)*ln(1+ x/x)/(x*lna)=lim ln(1+ x/x)A(x/ x)/(x*lna)=lim

5、 lne/(x*lna)=1/(x*lna)若 a=e，原函数 f(x)=IogeAx=Inx则 f(x)=1/(x*lne)=1/x( 6)f(x)=tanxf(x)=lim (tan(x+-nx)/ x=lim (sin(x+ x)/cos(x+nx/abxo/ x=lim (sin(x+ sinxcos x)Coscos(K+ x)/( xcos(x+ x)=lim (sin xcos xcosx+s inx+sinxsin x)/( xcos(x+ x)=lim sin x/( xcos(x+ x) =1/(cosx)A2=secx/cosx=(secx)A2=1+(tanx)A2 (

6、 7)f(x)=cotxf(x)=lim (cot(x+-tx)/ x=lim (cos(x+ x)/s in (xcosx/Anx)/ x=lim (cos(x+ x)sCosxs in(x+ x)/( xsin(x+ x)=lim (cWxs inxcosxsinxcos -coxsxsin xcosx)/( xsin(x+ x)=lim -sin x/( xsin(x+ x)=-1/(si nx)八2二-cscx/si nx=-(secx)八2=-1-(cotx)八2( 8)f(x)=secxf(x)=lim(sec(x+ -sexc)x)/ x=lim (1/cos(x+ -t/cx$

7、x)/ x=lim (cosx-cos(x+ x)/( xcos x) x/( xcc=lim (cosx-cosxcos x+sinxsin x)/( xcos(x+ x)=lim sinxsin二sin x/(cosx)A2=ta nx*secx( 9)f(x)=cscxf(x)=lim(csc(x+ -csxc)x)/ x=lim (1/sin(x+ -1 /sinxx) )/ x=lim (sinx-sin(x+ x)/( xsin(x+ x) x)=lim (sinx-sinxcos -xx xcosx)/( xsin(x+ -nx)Hmosx/( xsin(x+=-cosx/(si

8、nx)A2=-cotx*cscx( 10)f(x)=xAxlnf(x)=xlnx(lnf(x)=(xlnx)f(x)/f(x)=lnx+1f(x)=(lnx+1)*f(x)f(x)=(l nx+1)*xx(12)h(x)=f(x)g(x)h(x)=lim (f(x+ x)g(xl+x)g(x)/ Ax=lim (f(x+ A(x)+f(x)*g(x+ A x)+(ggx+-g(x+xA x)*f(x)/ Ax=lim (f(x+ -f(x)x*g(x+ A x)+(g(x+ Ax)-g(x)*f(x)+f(x)*g(x+ -f(x)*g(x+ A x)/ Ax=lim (f(x+ -f(xx*

9、g(x+ A x)/ A x+-g(x)*f(k)t) A x=f(x)g(x)+f(x)g(x)(13)h(x)=f(x)/g(x)h(x)=lim (f(x+ A x)/g(x+(x)g(xx)/ Ax=lim (f(x+ A x)6xXx+ A x)/( A xg(x)g(x+ A x)=lim (f(x+ -f(k)x)f(x)*g(x)-(g(x+ Ax)-g(x)+g(x)*f(x)/( A xg(x)g(x+ A x)=lim (f(x+ -f(k)x*g(x)-(g(x+ Ax)-g(x)*f(x)+f(x)g(x)f(x)g(x)/( A xg(x)g(x+ A x)=lim

10、 (f(x+ -f(x)*g(x)/( A xg(x)g-(g(x+ AAk)x)-g(x)*f(x)/( A xg(x)g(x+ A 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)x14)h(x)=f(g(x)h(x)lim f(g(x+-f(gx)/2xlim f(g(x+-g(x)+g(x)f(g(x)/ 2另 g(x)=u，g(x+ 2 x-g)(x)=2)ulim (f(u+-2f(uu)/ 2xlim (f(u+-2f(uu)* 2 u/(2 x* 2 u)lim f(u)*2 u/ 2

11、xlim f(u)*(g(x+ -g(x)/x)2x=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)=arcsinx则 siny=x(siny)=cosy所以(arcsinx)=1/(siny)=1/cosy=1/ Vis in yF2(siny=x)=1/ V-kA2即 f(x)=1/ VA2(16)y=f(x)=arctanx则 tany=x(tan y)=1+(ta ny)八2=1+xT所以(arcta nx)=+xT即 f(x)二 +xA2总结一

12、下( xAn)=nxA(n-1)( sinx)=cosx( cosx) =-si n x( aAx)=aAxlna( eAx)=eAx( logaAx)=1/(xlna)( lnx)=1/x(tanx)=(secx)A2=1+(tanx)A2 (cotx)=-(cscx)A2=-1-(cotx)A2 (secx)=tanx*secx(cscx)=-cotx*cscx(xAx)=(lnx+1)*xAx(arcsi nx)=1/ -x91(arctanx)=+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)