北师大版选修11 3.4.2 导数的乘法与除法法则 作业.docx

4.2导数的乘法与除法法则课时过关能力提升1.函数f(x)=exx的导数是()a.exx2b.ex(x-1)x2c.ex(1-x)x2d.ex(1+x)x2答案:b2.函数y=cosx1-x的导数是()a.-sinx+xsinx(1-x)2b.xsinx-sinx-cosx(1-x)2c.cosx-sinx+xsinx(1-x)2d.cosx-sinx+xsinx1-x解析:y=cosx1-x=(-sinx)(1-x)-cosx(-1)(1-x)2=cosx-sinx+xsinx(1-x)2.答案:c3. 若曲线f(x)=xsin x+1在x=2处的切线与直线ax+2y+1=0互相垂直,则实数a等于()a.-2b.-1c.1d.2答案:d4.函数y=(x+2a)(x-a)2的导数为()a.2(x2-a2)b.3(x2+a2)c.3(x2-a2)d.2(x2+a2)解析:y=(x+2a)(x-a)2+(x+2a)(x-a)2=(x-a)2+(x+2a)(x2-2ax+a2)=(x-a)2+2(x+2a)(x-a)=(x-a)(x-a+2x+4a)=3(x2-a2).答案:c5.函数y=2x2(x2+1)2的导数是()a.y=4x(x2+1)-8x3(x2+1)3b.y=4x(x2+1)-4x2(x2+1)3c.y=2x(x2+1)-8x3(x2+1)3d.y=4x(x2+1)-4x(x2+1)3答案:a6.已知y=xex,则y=.解析:y=(xex)=xex+x(ex)=ex+xex.答案:ex+xex7.已知函数f(x)=x(x-1)(x-2)(x-3)(x-4)(x-5),则f(0)=.解析:f(x)=(x-1)(x-2)(x-3)(x-4)(x-5)+x(x-1)(x-2)(x-3)(x-4)(x-5),f(0)=(-1)(-2)(-3)(-4)(-5)=-120.答案:-1208.求下列函数的导数:(1)y=3x2+xcos x;(2)y=lnxx2+1;(3)y=4-x3x2cosx;(4)y=1xcos x;(5)y=x5+x+sinxx2.解:(1)y=(3x2)+(xcos x)=6x+cos x-xsin x.(2)y=(lnx)(x2+1)-(lnx)(x2+1)(x2+1)2=1x(x2+1)-(lnx)2x(x2+1)2=x2+1-2x2lnxx(x2+1)2.(3)y=4-x3x2cosx=(4-x3)x2cosx-(4-x3)(x2cosx)(x2cosx)2=-3x2x2cosx-(4-x3)(2xcosx-x2sinx)x4cos2x=-x4cosx-8xcosx+4x2sinx-x5sinxx4cos2x=(4x-x4)sinx-(x3+8)cosxx3cos2x.(4)方法一:y=1xcosx=1xcos x+1x(cos x)=x-12cos x-1xsin x=-12x-32cos x-1xsin x=-cosx2x3-1xsin x=-cosx+2xsinx2xx.方法二:y=1xcosx=cosxx=(cosx)x-cosx(x)(x)2=-sinxx-cosx12x-12x=-xsinx+12xcosxx=-2xsinx+cosx2xx=-cosx+2xsinx2xx.(5)因为y=x3+x-32+sinxx2,所以y=(x3)+(x-32)+(sinx)x2-sinx(x2)x4=3x2-32x-52+x2cosx-2xsinxx4=3x2-32x-52+x-2cos x-2x-3sin x.9.已知函数f(x)=ax-6x2+b的图像在点m(-1,f(-1)处的切线的方程为x+2y+5=0,求函数的解析式.解:因为点m(-1,f(-1)在切线上,所以-1+2f(-1)+5=0,所以f(-1)=-2.因为f(x)=a(x2+b)-2x(ax-6)(x2+