《导数公式证明大全.docx》由会员分享，可在线阅读，更多相关《导数公式证明大全.docx（10页珍藏版）》请在三一文库上搜索。
1、导数的定义：f(x)=lim Ay/AxAx-0 (下面就不再标明Ax-0 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+ A x-x)(x+ A x)A(n-1)+x*(x+ A x)A(n-2)+.+xA(n-2)*(x+ A x)+xA(n-1)/A x=lim (x+ A x)A(n-1)+x*(x+ Ax)A(n-2)+.+xA(n-2)*(x+ A 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：
2、(n为任意实数)f3=xAnlnf(x)=nlnx(lnf(x)-(nlnx)'f(x)/f(x)=n/xf(x)=n/x*f(x) f(x)=n/x*xAn1)-f(x)=nxA(n2) f(x)=sinx f'(x)=lim (sin(x+ A x)-sinx)/ A x=lim (sinxcosA x+cosxsin A x-sinx)/ A x=lim (sinx+cosxsinA x-sinx)/ A x=lim cosxsinA x/A x=cosx( 3) f(x)=cosx f'(x)=lim (cos(x+ A x)-cosx)/A x=lim (co
3、sxcosA x-sinxsin A x-cosx)/A x=lim (cosx-sinxsin A x-cos)/A x=lim -sinxsinAx/Ax =-sinx(4) f(x)=aAx f(x)x A aAx)/-x) A (aA(x+ =lim=lim aAx*(aA A x-1)/ A x(设 aAAx-1=m,贝U 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)=l
4、im aAx*lna/lne=aAx*lna若a=e,原函数 f(x)=eAx则 f(x)=eAx*lne=eAx(5) f(x)=logaAxf(x)=lim (logaA(x+ A x)-logaAx)/ A x=lim logaA(1 +Ax/x)/ Ax=lim ln(1+ Ax/x)/(lna* Ax)x) A x/x)/(x*lna* A x*ln(1+ =lim=lim (x/ Ax)*ln(1+ A x/x)/(x*lna)=lim ln(1+ Ax/x)A(x/ Ax)/(x*lna)=lim lne/(x*lna)=1/(x*lna)若 a=e,原函数 f(x)=logeA
5、x=lnx则 f'(x)=1/(x*lne)=1/x( 6) f(x)=tanxf'(x)=lim (tan(x+ A x)-tanx)/ A x=lim (sin(x+ A x)/cos(x+ A x)-sinx/cosx)/Ax=lim (sin(x+ A x)cosx-sinxcos(x+ A x)/( A xcosxcos(x+A x)=lim (sinxcosA xcosx+sin A xcosxcosx-sinxcosxcosA x+sinxsinxsin A x)/( A xcosxcos(x+A x)=lim sin Ax/( A xcosxcos(x+A x)
6、=1/(cosx)A2=secx/cosx=(secx)A2=1+(tanx)A2( 7) f(x)=cotxf'(x) =lim (cot(x+ A x)-cotx)/ A x=lim (cos(x+Ax)/sin(x+ A x)-cosx/sinx)/Ax=lim (cos(x+ A x)sinx-cosxsin(x+ A x)/( A xsinxsin(x+ A x)=lim (cosxcosA xsinx-sinxsinxsin A x-cosxsinxcosA x-cosxsinA xcosx)/( A xsinxsin(x+ A x)=lim -sin A x/( A xs
7、inxsin(x+ A x)=-1/(sinx)A2=-cscx/sinx=-(secx)A2=-1-(cotx)A2( 8) f(x)=secxf'(x)=lim (sec(x+A x)-secx)/A x=lim (1/cos(x+Ax)-1/cosx)/A x=lim (cosx-cos(x+ A x)/( A xcosxcosA x)=lim (cosx-cosxcosA x+sinxsin A x)/( A xcosxcos(x+A x)=lim sinxsin A x/( A xcosxcos(x+A x)=sinx/(cosx)A2=tanx*secxf(x)=cscx
8、) 9(f'(x)=lim (csc(x+ A x)-cscx)/A x=lim (1/sin(x+ Ax)-1/sinx)/Ax=lim (sinx-sin(x+ Ax)/( Axsinxsin(x+ Ax)=lim (sinx-sinxcosA x-sin A xcosx)/( A xsinxsin(x+ A x)=lim -sin A xcosx/( A xsinxsin(x+ A x)=-cosx/(sinx)A2=-cotx*cscx10) f(x)=xAx lnf(x)=xlnx (lnf(x)'=(xlnx)' f'(x)/f(x)=lnx+1f&
9、#39;(x)=(lnx+1)*f(x) f(x)=(lnx+1)*xAx( 12) h(x)=f(x)g(x)h'(x)=lim (f(x+ A x)g(x+ A x)-f(x)g(x)/ A xx A x)*f(x)/ A g(x+-g(x)-x) A x)+(g(x+ A f(x)+f(x)*g(x+ -x) A (f(x+ =lim=lim (f(x+ A x)-f(x)*g(x+ A x)+(g(x+ A x)-g(x)*f(x)+f(x)*g(x+ Ax)-f(x)*g(x+ A x)/ A x=lim (f(x+ Ax)-f(x)*g(x+ Ax)/ Ax+(g(x+ A
10、x)-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+ A x)/( Axg(x)g(x+ Ax)=lim (f(x+ A x)-f(x)+f(x)*g(x) -(g(x+ Ax)-g(x)+g(x)*f(x)/( Axg(x)g(x+ Ax)=lim (f(x+ A x)-f(x)*g(x) -(g(x+Ax)-g(x)*f(x)+f(x)g(x) -f(x)g(x)/( Ax
11、g(x)g(x+A x)=lim (f(x+Ax)-f(x)*g(x)/( A xg(x)g(x+ A x)-(g(x+ A 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)x( 14) h(x)=f(g(x)h'(x)=lim f(g(x+ Ax)-f(g(x)/ Axx A f(g(x)/ -g(x)+g(x) -x) A f(g(x+ =lim(另 g(x)=u, g(x+Ax)-g(x
12、)= Au)=lim (f(u+ A u)-f(u)/ A x=lim (f(u+ A u)-f(u)* A u/( A x* A u)=lim f(u)* A u/A x=lim f(u)*(g(x+ Ax)-g(x)/ Ax=f'(u)*g'(x)=f'(g(x)g'(x)总结一下(xAn )'二nxA(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)'=tanx*secx(cscx)'=-cotx*cscx(xAx)'=(lnx+1)*xAxf(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)