同濟版高等數(shù)學-函數(shù)的求導法則課件

第二節(jié)函數(shù)的求導法則和、差、積、商的求導法則反函數(shù)的導數(shù)三、復合函數(shù)的求導法則四、初等函數(shù)的求導問題五、小結及作北第二節(jié)函數(shù)的求導法則1導數(shù)概念的回顧導數(shù)的定義f(x)=limf(x+△x)-f△y2、導數(shù)幾何意義f(x0)表示曲線y=f(x)在點M(x,f(x0)處的切線的斜率。3、求導公式(C)=0(sinx)=cosx(cosx)sInr導數(shù)概念的回顧2(x)=x(∈R(ar)=aa(logx)=rInanx(x)=x(∈R3一、和、差、權、商的求導法則定理如果函數(shù)u(x),v(x)在點x處可導,則它們的和、差、積、商(分母不為零)在點x處也可導,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、商的求導法則4證(1)、(2)略證(3)u(x)u'(xv(x-u(x)v'(x(v(x)≠0設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,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的導數(shù)解y′=3x2-4x+cosx例2求y=sin2xlmx的導數(shù)解:因為y=2sinx.cosx·lnx所以y=2cosx:cosx·lnx+2sinx(-sinx)lnx+2sinx·cOsx2coS2xlnx+-sin2x例1求y=x3-2x2+six+cos的導數(shù)8例3求y=tanx的導數(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的導數(shù)9例4求y=secx的導數(shù)解y=(secx)=/(cosx)sinx=secrtanrcosxcosr因此(secx)=secxtanx同理可得cscx)=-cscrcotx.例4求y=secx的導數(shù)10同濟版高等數(shù)學-函數(shù)的求導法則課件11同濟版高等數(shù)學-函數(shù)的求導法則課件12同濟版高等數(shù)學-函數(shù)的求導法則課件13同濟版高等數(shù)學-函數(shù)的求導法則課件14同濟版高等數(shù)學-函數(shù)的求導法則課件15同濟版高等數(shù)學-函數(shù)的求導法則課件16同濟版高等數(shù)學-函數(shù)的求導法則課件17同濟版高等數(shù)學-函數(shù)的求導法則課件18同濟版高等數(shù)學-函數(shù)的求導法則課件19同濟版高等數(shù)學-函數(shù)的求導法則課件20同濟版高等數(shù)學-函數(shù)的求導法則課件21同濟版高等數(shù)學-函數(shù)的求導法則課件22同濟版高等數(shù)學-函數(shù)的求導法則課件23同濟版高等數(shù)學-函數(shù)的求導法則課件24同濟版高等數(shù)學-函數(shù)的求導法則課件25同濟版高等數(shù)學-函數(shù)的求導法則課件26同濟版高等數(shù)學-函數(shù)的求導法則課件27同濟版高等數(shù)學-函數(shù)的求導法則課件28同濟版高等數(shù)學-函數(shù)的求導法則課件29同濟版高等數(shù)學-函數(shù)的求導法則課件30同濟版高等數(shù)學-函數(shù)的求導法則課件31同濟版高等數(shù)學-函數(shù)的求導法則課件32同濟版高等數(shù)學-函數(shù)的求導法則課件33同濟版高等數(shù)學-函數(shù)的求導法則課件34同濟版高等數(shù)學-函數(shù)的求導法則課件35同濟版高等數(shù)學-函數(shù)的求導法則課件36同濟版高等數(shù)學-函數(shù)的求導法則課件37同濟版高等數(shù)學-函數(shù)的求導法則課件38同濟版高等數(shù)學-函數(shù)的求導法則課件39同濟版高等數(shù)學-函數(shù)的求導法則課件40同濟版高等數(shù)學-函數(shù)的求導法則課件41同濟版高等數(shù)學-函數(shù)的求導法則課件42同濟版高等數(shù)學-函數(shù)的求導法則課件43同濟版高等數(shù)學-函數(shù)的求導法則課件44同濟版高等數(shù)學-函數(shù)的求導法則課件45同濟版高等數(shù)學-函數(shù)的求導法則課件46同濟版高等數(shù)學-函數(shù)的求導法則課件47同濟版高等數(shù)學-函數(shù)的求導法則課件48同濟版高等數(shù)學-函數(shù)的求導法則課件49同濟版高等數(shù)學-函數(shù)的求導法則課件50同濟版高等數(shù)學-函數(shù)的求導法則課件51同濟版高等數(shù)學-函數(shù)的求導法則課件52同濟版高等數(shù)學-函數(shù)的求導法則課件53同濟版高等數(shù)學-函數(shù)的求導法則課件54同濟版高等數(shù)學-函數(shù)的求導法則課件55第二節(jié)函數(shù)的求導法則和、差、積、商的求導法則反函數(shù)的導數(shù)三、復合函數(shù)的求導法則四、初等函數(shù)的求導問題五、小結及作北第二節(jié)函數(shù)的求導法則56導數(shù)概念的回顧導數(shù)的定義f(x)=limf(x+△x)-f△y2、導數(shù)幾何意義f(x0)表示曲線y=f(x)在點M(x,f(x0)處的切線的斜率。3、求導公式(C)=0(sinx)=cosx(cosx)sInr導數(shù)概念的回顧57(x)=x(∈R(ar)=aa(logx)=rInanx(x)=x(∈R58一、和、差、權、商的求導法則定理如果函數(shù)u(x),v(x)在點x處可導,則它們的和、差、積、商(分母不為零)在點x處也可導,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、商的求導法則59證(1)、(2)略證(3)u(x)u'(xv(x-u(x)v'(x(v(x)≠0設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,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的導數(shù)解y′=3x2-4x+cosx例2求y=sin2xlmx的導數(shù)解:因為y=2sinx.cosx·lnx所以y=2cosx:cosx·lnx+2sinx(-sinx)lnx+2sinx·cOsx2coS2xlnx+-sin2x例1求y=x3-2x2+six+cos的導數(shù)63例3求y=tanx的導數(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的導數(shù)64例4求y=secx的導數(shù)解y=(secx)=/(cosx)sinx=secrtanrcosxcosr因此(secx)=secxtanx同理可得cscx)=-cscrcotx.例4求y=secx的導數(shù)65同濟版高等數(shù)學-函數(shù)的求導法則課件66同濟版高等數(shù)學-函數(shù)的求導法則課件67同濟版高等數(shù)學-函數(shù)的求導法則課件68同濟版高等數(shù)學-函數(shù)的求導法則課件69同濟版高等數(shù)學-函數(shù)的求導法則課件70同濟版高等數(shù)學-函數(shù)的求導法則課件71同濟版高等數(shù)學-函數(shù)的求導法則課件72同濟版高等數(shù)學-函數(shù)的求導法則課件73同濟版高等數(shù)學-函數(shù)的求導法則課件74同濟版高等數(shù)學-函數(shù)的求導法則課件75同濟版高等數(shù)學-函數(shù)的求導法則課件76同濟版高等數(shù)學-函數(shù)的求導法則課件77同濟版高等數(shù)學-函數(shù)的求導法則課件78同濟版高等數(shù)學-函數(shù)的求導法則課件79同濟版高等數(shù)學-函數(shù)的求導法則課件80同濟版高等數(shù)學-函數(shù)的求導法則課件81同濟版高等數(shù)學-函數(shù)的求導法則課件82同濟版高等數(shù)學-函數(shù)的求導法則課件83同濟版高等數(shù)學-函數(shù)的求導法則課件84同濟版高等數(shù)學-函數(shù)的求導法則課件85同濟版高等數(shù)學-函數(shù)的求導法則課件86同濟版高等數(shù)學-函數(shù)的求導法則課件87同濟版高等數(shù)學-函數(shù)的求導法則課件88同濟版高等數(shù)學-函數(shù)的求導法則課件89同濟版高等數(shù)學-函數(shù)的求導法則課件90同濟版高等數(shù)學-函數(shù)的求導法則課件91同濟版高等數(shù)學-函數(shù)的求導法則課件92同濟版高等數(shù)學-函數(shù)的求導法則課件93同濟版高等數(shù)學-函數(shù)的求導法則課件94同濟版高等數(shù)學-函數(shù)的求導法則課件95同濟版高等數(shù)學-函數(shù)的求導法則課件96同濟版高等數(shù)學-函數(shù)的求導法則課件97同濟版高等數(shù)學-函數(shù)的求導法則課件98同濟版高等數(shù)學-函數(shù)的求導法則課件99同濟版高等數(shù)學-函