同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件

第二節(jié)函數(shù)的求導(dǎo)法則和、差、積、商的求導(dǎo)法則反函數(shù)的導(dǎo)數(shù)三、復(fù)合函數(shù)的求導(dǎo)法則四、初等函數(shù)的求導(dǎo)問題五、小結(jié)及作北第二節(jié)函數(shù)的求導(dǎo)法則1導(dǎo)數(shù)概念的回顧導(dǎo)數(shù)的定義f(x)=limf(x+△x)-f△y2、導(dǎo)數(shù)幾何意義f(x0)表示曲線y=f(x)在點(diǎn)M(x,f(x0)處的切線的斜率。3、求導(dǎo)公式(C)=0(sinx)=cosx(cosx)sInr導(dǎo)數(shù)概念的回顧2(x)=x(∈R(ar)=aa(logx)=rInanx(x)=x(∈R3一、和、差、權(quán)、商的求導(dǎo)法則定理如果函數(shù)u(x),v(x)在點(diǎn)x處可導(dǎo),則它們的和、差、積、商(分母不為零)在點(diǎn)x處也可導(dǎo),并且(1)[u(x)±v(x)y=l(x)±v(x);(2)[(x)·v(x)=l(x)w(x)+l(x)v(x);u(xu'(xv(r-u(xv(x)(v(x)≠0)一、和、差、權(quán)、商的求導(dǎo)法則4證(1)、(2)略證(3)u(x)u'(xv(x-u(x)v'(x(v(x)≠0設(shè)f(x)u(xD(,(v(x)≠0),f(x)=lif(x+h)-∫(x)h→0h(x+h)u(x)=limv(x+h)v(r)lipu(x+h)v(x)-u(x)v(x+h)v(x+hv(x)h證(1)、(2)略5lu(x+h)-u(xlv(x-u(xv(x+h)-v(x=mh→0v(x+hv(x)hu(x+hj-u(x)v(x)-(4)、D(x+h)-v(x)h→0v(r+hv(x)u'(x)(x)-l(x)’(x)所以f(x)在x處可導(dǎo),且有u(x)u(xv(r-u(xv(x(v(x)≠0lu(x+h)-u(xlv(x-u(xv(x+h)-v(x6推論(1)∑f(xy=∑fx);(2)[Cf(x)=Cf(x)If(x=f(x)f2(x)…f(+f1(x)f2(x)…f(x)+…+f1(x)f2(x)…f(x)∑∏f(x)f(x;*H(f(x)f,(x)f(x))'=f(x)5(x)f(x+f1(x)2(x)3(x)+f1(x)2(x)3(x)推論7例1求y=x3-2x2+six+cos的導(dǎo)數(shù)解y′=3x2-4x+cosx例2求y=sin2xlmx的導(dǎo)數(shù)解:因?yàn)閥=2sinx.cosx·lnx所以y=2cosx:cosx·lnx+2sinx(-sinx)lnx+2sinx·cOsx2coS2xlnx+-sin2x例1求y=x3-2x2+six+cos的導(dǎo)數(shù)8例3求y=tanx的導(dǎo)數(shù)解y=(anx)=sInr(sinx)'cosx-sinx(cosx)cosCcosr+sinrseccoscosx因此(tanx)secr同理可得(cot)'=SInx例3求y=tanx的導(dǎo)數(shù)9例4求y=secx的導(dǎo)數(shù)解y=(secx)=/(cosx)sinx=secrtanrcosxcosr因此(secx)=secxtanx同理可得cscx)=-cscrcotx.例4求y=secx的導(dǎo)數(shù)10同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件11同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件12同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件13同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件14同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件15同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件16同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件17同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件18同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件19同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件20同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件21同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件22同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件23同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件24同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件25同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件26同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件27同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件28同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件29同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件30同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件31同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件32同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件33同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件34同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件35同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件36同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件37同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件38同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件39同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件40同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件41同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件42同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件43同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件44同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件45同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件46同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件47同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件48同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件49同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件50同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件51同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件52同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件53同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件54同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件55第二節(jié)函數(shù)的求導(dǎo)法則和、差、積、商的求導(dǎo)法則反函數(shù)的導(dǎo)數(shù)三、復(fù)合函數(shù)的求導(dǎo)法則四、初等函數(shù)的求導(dǎo)問題五、小結(jié)及作北第二節(jié)函數(shù)的求導(dǎo)法則56導(dǎo)數(shù)概念的回顧導(dǎo)數(shù)的定義f(x)=limf(x+△x)-f△y2、導(dǎo)數(shù)幾何意義f(x0)表示曲線y=f(x)在點(diǎn)M(x,f(x0)處的切線的斜率。3、求導(dǎo)公式(C)=0(sinx)=cosx(cosx)sInr導(dǎo)數(shù)概念的回顧57(x)=x(∈R(ar)=aa(logx)=rInanx(x)=x(∈R58一、和、差、權(quán)、商的求導(dǎo)法則定理如果函數(shù)u(x),v(x)在點(diǎn)x處可導(dǎo),則它們的和、差、積、商(分母不為零)在點(diǎn)x處也可導(dǎo),并且(1)[u(x)±v(x)y=l(x)±v(x);(2)[(x)·v(x)=l(x)w(x)+l(x)v(x);u(xu'(xv(r-u(xv(x)(v(x)≠0)一、和、差、權(quán)、商的求導(dǎo)法則59證(1)、(2)略證(3)u(x)u'(xv(x-u(x)v'(x(v(x)≠0設(shè)f(x)u(xD(,(v(x)≠0),f(x)=lif(x+h)-∫(x)h→0h(x+h)u(x)=limv(x+h)v(r)lipu(x+h)v(x)-u(x)v(x+h)v(x+hv(x)h證(1)、(2)略60lu(x+h)-u(xlv(x-u(xv(x+h)-v(x=mh→0v(x+hv(x)hu(x+hj-u(x)v(x)-(4)、D(x+h)-v(x)h→0v(r+hv(x)u'(x)(x)-l(x)’(x)所以f(x)在x處可導(dǎo),且有u(x)u(xv(r-u(xv(x(v(x)≠0lu(x+h)-u(xlv(x-u(xv(x+h)-v(x61推論(1)∑f(xy=∑fx);(2)[Cf(x)=Cf(x)If(x=f(x)f2(x)…f(+f1(x)f2(x)…f(x)+…+f1(x)f2(x)…f(x)∑∏f(x)f(x;*H(f(x)f,(x)f(x))'=f(x)5(x)f(x+f1(x)2(x)3(x)+f1(x)2(x)3(x)推論62例1求y=x3-2x2+six+cos的導(dǎo)數(shù)解y′=3x2-4x+cosx例2求y=sin2xlmx的導(dǎo)數(shù)解:因?yàn)閥=2sinx.cosx·lnx所以y=2cosx:cosx·lnx+2sinx(-sinx)lnx+2sinx·cOsx2coS2xlnx+-sin2x例1求y=x3-2x2+six+cos的導(dǎo)數(shù)63例3求y=tanx的導(dǎo)數(shù)解y=(anx)=sInr(sinx)'cosx-sinx(cosx)cosCcosr+sinrseccoscosx因此(tanx)secr同理可得(cot)'=SInx例3求y=tanx的導(dǎo)數(shù)64例4求y=secx的導(dǎo)數(shù)解y=(secx)=/(cosx)sinx=secrtanrcosxcosr因此(secx)=secxtanx同理可得cscx)=-cscrcotx.例4求y=secx的導(dǎo)數(shù)65同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件66同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件67同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件68同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件69同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件70同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件71同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件72同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件73同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件74同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件75同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件76同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件77同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件78同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件79同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件80同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件81同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件82同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件83同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件84同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件85同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件86同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件87同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件88同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件89同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件90同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件91同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件92同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件93同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件94同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件95同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件96同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件97同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件98同濟(jì)版高等數(shù)學(xué)-函數(shù)的求導(dǎo)法則課件99同濟(jì)版高等數(shù)學(xué)-函