导数公式证明大全

1、学习好资料欢迎下载导数的定义：f(x)=lim y/ x x70 （下面就不再标明厶XT0 了）=lim -sin xcosx/( xsinxsin(x+ x)=-cosx/(sinx)2=-cotx*cscx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+ x)A n-xn/ x x)+xA( n-1)/ x=lim (x+ x-x)(x+ x)A( n-1)+x*(x+ x)( n-2)+.+xA( n-2)*(x+ x)+xA( n-1)=lim (x+ x)A( n-1)+x*(x+ x)( n-2)+.+xA( n-2)*(x+ =xA( n

2、-1)+x*xA( n-2)+xA2*xA( n-3)+ .x( n-2)*x+xA( n-1)=nxA( n-1)证法二：（n为任意实数）f(x)=xA nIn f(x)=ninx(I nf(x)'=( ninx)'f(x)/f(x)=n/x f(x)=n/x*f(x) f(x)=n/x*xA nf(x)=n-1)(2) f(x)=sinxf(x)=lim (sin(x+ x)-si nx)/ x=lim (sin xcos x+cosxsin x-sinx)/ x=lim (sin x+cosxs in x-si nx)/ x=lim cosxs in x/ x=cosx(

3、3) f(x)=cosxf(x)=lim (cos(x+ x)-cosx)/ x=lim (cosxcos x-s in xsin x-cosx)/ x=lim (cosx-s inxsin x-cos)/ x=lim -si nxsin x/ x=-s inx(4) f(x)=axf(x)=lim (aA(x+ x)-ax)/ x=lim ax*(a x-1)/ x(设 x-1=m,则厶 x=loga(m+1)=lim ax*m/loga(m+1)=lim ax*m/l n( m+1)/l na=lim ax*l na*m/l n(m+1) =lim ax*l na/(1/m)* In (m

4、+1)=lim ax*l na/In (m+1)A(1/m)=lim ax* Ina/ln e=ax* Ina若 a=e,原函数 f(x)=ex 贝U f'(x)=ex*Ine=ex(5) f(x)=logaxf(x)=lim (loga(x+ x)-logax)/ x=lim loga(x+ x)/x/ x=lim loga(1+ x/x)/ x=lim ln (1+ x/x)/(l na* X)=lim x*l n(1+ x/x)/(x*l na* x)=lim (x/ x)*l n(1+ x/x)/(x*l na)=lim ln (1+ x/x)A(x/ x)/(x*l na)=

5、lim ln e/(x*l na)=1/(x*l na)若 a=e，原函数 f(x)=logex=lnx贝U f(x)=1/(x*lne)=1/x(6) f(x)=ta nxf(x)=lim (ta n(x+ x)-ta nx)/ x=lim (si n(x+ x)/cos(x+ x)-si nx/cosx)/=lim (si n(x+ x)cosx-s in xcos(x+ x)/( xcosxcos(x+ x)=lim (sinxcos xcosx+sin xcosxcosx-sinxcosxcos x+sinxsinxsin x)/(xcosxcos(x+ x)=lim sin x/(

6、xcosxcos(x+ x)=1/(cosx)2=secx/cosx=(secx)2=1+(tanx)A2(7) f(x)=cotxf(x)=lim (cot(x+ x)-cotx)/ x=lim (cos(x+ x)/si n(x+ x)-cosx/si nx)/=lim (cos(x+ x)si nx-cosxsi n(x+ x)/( xsinxsin(x+ x)=lim (cosxcos xsi nx-si nxsi nxsi n x-cosxsi nxcos x-cosxsi n xcosx)/(xsinxsin(x+ x)=lim -sin x/( xsinxsin(x+ x)=-1

7、/(sinx)2=-cscx/sinx=-(secx)2=-1-(cotx)2(8) f(x)=secx f(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=lim si nxsin x/( x+sinxsin x)/( xcosxcos(x+ x) xcosxcos(x+ x)=si nx/(cosx)A2=ta nx*secx(9) f(x)=cscxf(x)=lim(csc(x+ x)-cscx)/ x=lim (1/si n

8、(x+ x)-1/si nx)/ x=lim (si nx-si n(x+ x)/( xsinxsin(x+ x)=lim (sinx-sinxcos x-sin xcosx)/( xsinxsin(x+ x)学习好资料欢迎下载(10) f(x)=xAxlnf(x)=xlnx(Inf(x)'=(xlnx)'f(x)/f(x)=l nx+1f(x)=(l nx+1)*f(x)f(x)=(l nx+1)*xAx(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(x)+f(x)*g(x

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

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

11、g'(x)/(g(x)*g(x)=f(x)g(x)-f(x)g'(x)/(g(x)*g(x)x x)-f(g(x)/ x(14) h(x)=f(g(x) h'(x)=lim f(g(x+=lim f(g(x+ x)-g(x)+g(x)-f(g(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 ) '=nxA(n-1)(sinx ) '=cosx(cosx) '=-sinx(ax) '=axIna(ex) '=eAx (logax ) '=1/(x1 na)(lnx ) '=1/