导数公式证明大全

1、导数的定义：f(x)=lim y/ x x0 (下面就不再标明 x0 了)用定义求导数公式(1) f(x)=xAn证法一：( n 为自然数)f'(x)=lim (x+ x)An-xAn/ x=lim (x+ x-x)(x+ x)A(n-1)+x*(x+ x)A(n-2)+.+xA(n-2)*(x+ x)+xA(n-1)/ x=lim (x+ x)A(n-1)+x*(x+ x)A(n-2)+.+xA(n-2)*(x+ x)+xA(n-1) =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 为任意实数)f(

2、x)=xAnlnf(x)=nlnx(lnf(x)'=(nlnx)'f'(x)/f(x)=n/x f'(x)=n/x*f(x)f(x)=n/x*x5f(x)=nx( n-1)(2) f(x)=sinxf'(x)=lim (sin(x+ x)-sinx)/ x=lim (sinxcos x+cosxsin x-sinx)/ x=lim (sinx+cosxsin x-sinx)/ x=lim cosxsin x/ x=cosx(3) f(x)=cosxf'(x)=lim (cos(x+ x)-cosx)/ x=lim (cosxcos x-sinxs

3、in x-cosx)/ x=lim (cosx-sinxsin x-cos)/ x=lim -sinxsin x/ x=-sinx(4) f(x)=aAxf'(x)=lim (aA(x+ x)-aAx)/ x=lim aAx*(aA x-1)/ x(设 aA x-1=m，贝U x=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*l na若a

4、=e，原函数f(x)=eAx则 f'(x)=eAx*lne=eAx(5) f(x)=logaAxf'(x)=lim (logaA(x+ x)-logaAx)/ x=lim logaA(x+ x)/x/ x=lim logaA(1+ x/x)/ x=lim ln(1+ x/x)/(lna* x)=lim x*ln(1+ x/x)/(x*Ina* x)=lim (x/ x)*ln(1+ x/x)/(x*Ina)=lim ln(1+ x/x)A(x/ x)/(x*lna)=lim lne/(x*lna)=1/(x*lna)若 a=e，原函数 f(x)=logeAx=Inx则 f

5、9;(x)=1/(x*lne)=1/x(6) f(x)=tanxf'(x)=lim (tan(x+ x)-tanx)/ x=lim (sin(x+ x)/cos(x+ x)-sinx/cosx)/ x=lim (sin(x+ x)cosx-sinxcos(x+ x)/( xcosxcos(x+ x)=lim(sinxcos xcosx+sin xcosxcosx-sinxcosxcos x+sinxsinxsin x)/( xcosxco s(x+ x)=lim sin x/( xcosxcos(x+ x)=1/(cosx)A2=secx/cosx=(secx)A2=1+(tanx)A

6、2(7) f(x)=cotx f'(x)=lim (cot(x+ x)-cotx)/ x=lim (cos(x+ x)/sin(x+ x)-cosx/sinx)/ x=lim (cos(x+ x)sinx-cosxsin(x+ x)/( xsinxsin(x+ x)=lim(cosxcos xsinx-sinxsinxsin x-cosxsinxcos x-cosxsin xcosx)/( xsinxsin( x+ x)=lim -sin x/( xsinxsin(x+ x)=-1/(si nxF2=-cscx/si nx=-(secxF2=-1-(cotxF2(8) f(x)=sec

7、xf'(x)=lim (sec(x+ x)-secx)/ x=lim (1/cos(x+ x)-1/cosx)/ x=lim (cosx-cos(x+ x)/( xcosxcos x)=lim (cosx-cosxcos x+sinxsin x)/( xcosxcos(x+ x)=lim sinxsin x/( xcosxcos(x+ x)=sin x/(cosxF2=ta nx*secx(9) f(x)=cscxf'(x)=lim (csc(x+ x)-cscx)/ x=lim (1/sin(x+ x)-1/sinx)/ x=lim (sinx-sin(x+ x)/( xsi

8、nxsin(x+ x)=lim (sinx-sinxcos x-sin xcosx)/( xsinxsin(x+ x)=lim -sin xcosx/( xsinxsin(x+ x)=-cosx/(s in x)A2=-cotx*cscx(10 ) f(X)=xAxlnf(x)=xlnx(lnf(x)'=(xlnx)'f'(x)/f(x)=lnx+1 f'(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(x+ x)-f(x)g(x)/ x=lim (f(x+ x)-f

9、(x)+f(x)*g(x+ x)+(g(x+ x)-g(x)-g(x+ x)*f(x)/ x=lim(f(x+ x)-f(x)*g(x+ x)+(g(x+ x)-g(x)*f(x)+f(x)*g(x+ x)-f(x)*g(x+ x)/ x=lim (f(x+ x)-f(x)*g(x+ x)/ x+(g(x+ x)-g(x)*f(x)/ x=f'(x)g(x)+f(x)g'(x)( 13)h(x)=f(x)/g(x)h'(x)=lim (f(x+ x)/g(x+ x)-f(x)g(x)/ x=lim (f(x+ x)g(x)-f(x)g(x+ x)/( xg(x)g(x+

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

11、)*g(x)x(14)h(x)=f(g(x)h'(x)=lim f(g(x+ x)-f(g(x)/ x=lim f(g(x+ x)-g(x)+g(x)-f(g(x)/ x(另 g(x)=u , g(x+ x)-g(x)= u)=lim (f(u+ u)-f(u)/ x=lim (f(u+ u)-f(u)* u/( x* u)=lim f(u)* u/ x=lim f(u)*(g(x+ x)-g(x)/ x=f'(u)*g'(x)=f'(g(x)g'(x)总结一下(xAn) '=nx(n-1)( sinx )'=cosx( cosx)'=-sinx( aAx)'=aAxlna( eAx)'=eAx( logaAx )'=1/(xlna)( lnx) '=1/x(tanx)'=(secx)A2=1+(tanx)A2(cotx)'=-(cscx)A2=-1-(cotx)A2(secx)'