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1、Chapter3Derivatives(continued)BasicElementaryFunctions1.c0aa12.(x)ax,aR11x3.(logx)aInparticular,ifae,then(lnx)xlna4.(a)alnaInparticular,ifae,then(e)exxxx5.1)(sinx)cosx2)(cosx)sinx223)(tanx)secx4)(cotx)cscx5)(secx)tanxsecx6)(cscx)cotxcscx116.1)(arcsinx),1x12)(arccosx)4)(arccotx),1x11x21x211x211x23)(a
2、rctanx),xR,xRf(xh)f(x)hcclimProof1.f(x)limlim00h0h0h0hf(xh)f(x)hlog(xh)logx1xhxaa3.f(x)limlimlimlogah0h0hh0h1x1xhh1xhx1x1limlog(1)hlimlog(1)hxlimlog(10 x)hlogeaaaah0 xhxxhxlna01xxxylnaalna4.Letya,thenxlogy(a)a(logy)ah2xh22sincos2f(xh)f(x)5.1)f(x)limsin(xh)sinxhlimlimcosxh0hh0h0hxycosxyNotesinxsiny2s
3、in2)Likewise(同理可证)22223)(sinxcosx)cosxsinx2secx4)Likewise2cosx10sinx5)()tanxsecx6)Likewise2cosxcosx6.1)Letyarcsinx,thenxsiny1111(arcsinx)(siny)cosy21x21siny2)Likewise11113)Letyarctanx,thenxtany(arctanx)2secy1tany1x22(tany)4)LikewiseCombinations(Addition,Subtraction,Multiplication,Division)1.(fg)fg2.
4、(fg)fgfgInparticular,(cg)cgNote(fgh)(fg)h(fg)h(fg)h(fgfg)h(fg)hfghfghfgh(ff21f)nf1f2fnff21fnf1f2fnf3.()gfgfg,(g0)g2Proof1.LetFfg,thenF(xh)F(x)f(xh)g(xh)f(x)g(x)limh0hFlimh0hf(xh)f(x)g(xh)g(x)limlimfgh0hh0h2.LetFfg,thenF(xh)F(x)f(xh)g(xh)f(x)g(x)limh0hFlimh0hg(xh)f(xh)f(x)f(x)g(xh)g(x)limh0hlimg(xh)f
5、(xh)f(x)f(x)limg(xh)g(x)h0hh0hf(xh)f(x)limg(xh)limf(x)g(x)h0h0hg(x)f(x)f(x)g(x)Noteg(x)isdifferentiableg(x)iscontinuouslimg(xh)g(x)h03.LetFf,thengf(xh)f(x)g(xh)g(x)hF(xh)F(x)hf(xh)g(x)f(x)g(xh)limh0g(x)g(xh)hFlimlimh0h0g(x)f(xh)f(x)f(x)g(x)f(x)g(xh)0g(x)g(xh)hlimhg(x)f(xh)f(x)f(x)g(xh)g(x)hhlimh0g(x
6、)g(xh)f(xh)f(x)hg(xh)g(x)f(x)limhh0g(x)limh0g(x)limg(xh)h0g(x)f(x)f(x)g(x)g(x)g(x)InversefunctionsTheoremIf(1)yf(x)isdifferentiableatxandf(x)0,00(2)yf(x)iscontinuousandstrictlyincreasingdecreasingonaneighborhoodofx,0thenitsinversefunctionx(y)isdifferentiableaty,where01.f(x0)y0f(x)and(y0)0 x11Proof1(
7、y)limlimx00y0yyxf(x0)(y)(y0)yy0 xx0.f(x)f(x)f(x0)1Proof2(y)limlim0yyyy000Exercises2x2x1.1)y4xsin1(y44x3)3x2xx2)y5x23e(y15x2ln23e)3233)yxcosx(y3xcosxxsinx)324)ytanxsecx(ysecxtanxsecx)3225)yxlnx(y3xlnxx)exx2(x2)ex)x36)y7)yln3(yx1(yx12)(x1)2228)yxlnxcosx(y2xlnxcosxxcosxxlnxsinx)2ecsc)9)ecot(1)ecot22arc
8、sinv(uarctanv(1v)arctanv1varcsinv)10)u222(1v)1v(arctanv)553)22.1)Lety2sinx5cosx.Findy(3)andy(1x26x313dd62)62)Lettansin.Find(2431x3)Letf(x).Findf(0)(1)andf(2)(5)1x323.Findanequationofthetangentlinetoyxx2suchthatitsparallel(平行)tox+y-3=0.(Solutionx+y+3=0)Chapter3Derivatives(continued)1.theChainRule(链式法
9、那么)dydydudxdudxyF(x)f(g(x)yf(g(x)g(x)or2Example1FindF(x)ifF(x)x1.111212x2(x1)2(x1)2(x1)Solution1F(x)222x2x111dFdFdu1x2Solution2Letux1,thenF(x)u22u2xdxdudx22x1ExercisesExample21)Findyifylnf(x).1Solutionyf(x)f(x)Example3logarithmicdifferentiation1)Findyifyxsinx11ycosxlnx(sinx)xSolutionlnysinxlnxysinx)
10、xsinxyx(cosxlnx32x1(3x2)5x42)Findyify3412Solutionlnylnxln(x1)5ln(3x2)231y3112x3xx213x15)(3x2)4xx13x24y5y(2524x2x13x22.Implicit(隐函数)DifferentiationyExample1Findyifexye0.yxeyyySolutioneyyxy00ywhereysatisfiesexye0.2Example2Findanequationofthetangenttothecirclexy225at(3,4).122xxSolution1y25x2y25x225x233
11、4y|x325934y4(x3)or3x4y25xyxSolution22x2yy0y25x23.HigherDerivatives2nf(xh)f(x)ddy()dxdxdydy(n)y(y)lim,f(x)dx2h0hdxnExample1Find(cosx)(27).Solution(cosx)sinx,(cosx)cosx,(cosx)sinx,(cosx)(4)cosx(cosx)(24)cosx,(cosx)(27)sinxExample2Ifyx6x5x3,findy(n).3226,y(n)0foralln4Solutiony3x12x5,y6x12,y1(n)Example3
12、Iff(x),findf(x).xSolutionf(x)x1f(x)x,f(x)(1)(2)x,f(x)(1)(2)(3)x423,f(x)(1)n!x(n1)(n)n44Example4Findyifxy16.x3y333x3y3Solution4x4yy0y(3xy33xyy)3xy33xy(xy3)23423433x(y3xy7)24y3xy(yx4)27448xy72Exercises2221.1)ycot(x);(y2xcscx)xe)x2)yarctan(e);(y1e2x2arcsinx)1x223)y(arcsinx);(y12.1)yarccos1(2x);(y)xx212
13、22)yln(xxa);(y)xa223)yln(cscxcotx);(ycscx)2arccosxx2)24)y(arccos);(y24x2222223.1)ycosxcos(x);(y(sin2xcosx2xcosxsinx)4arccotxx2)22)y(arccot);(y24x23)ysec;(ysectanx)x2x2222dy;(exyy)xy4.Letxye.Finddxxexy(2x1)2323x;(y(2x1)2323x4125.1)y)2x123x3(x3)(x3)2(x3)233sinxsinx2)y(cosx);(y(cosx)cosxlncosxtanxsinx)