Monday, March 28, 2016

Single Variable Calculus, Chapter 7, 7.4-1, Section 7.4-1, Problem 56

Determine d9dx9(x8lnx)

ddx(x8lnx)=x8ddx(lnx)+(lnx)ddx(x8)ddx(x8lnx)=x81x+lnx8x7ddx(x8lnx)=x7+8x7lnx


d2dx2(x8lnx)=ddx(x7)+[8x7ddx(lnx)+(lnx)8ddx(x7)]d2dx2(x8lnx)=7x6+8x71x+lnx56x6d2dx2(x8lnx)=7x6+8x6+56x6lnxd2dx2(x8lnx)=15x6+56x6lnx


d3dx3(x8lnx)=15ddx(x6)+[56x6ddx(lnx)+(lnx)56ddx(x6)]d3dx3(x8lnx)=15(6x5)+56x61x+lnx56(6x5)d3dx3(x8lnx)=90x5+56x5+336x5lnxd3dx3(x8lnx)=146x5+336x5lnx



d4dx4(x8lnx)=146ddx(x5)+[336x5ddx(lnx)+(lnx)336ddx(x5)]d4dx4(x8lnx)=146(5x4)+336x51x+lnx336(5x4)d4dx4(x8lnx)=730x4+336x4+1680x4lnxd4dx4(x8lnx)=1066x4+1680x4lnx



d5dx5(x8lnx)=1066ddx(x5)+[1680x4ddx(lnx)+(lnx)1680ddx(x4)]d5dx5(x8lnx)=1066(4x3)+1680x41x+lnx1680(4x3)d5dx5(x8lnx)=4264x3+1680x3+6720x3lnxd5dx5(x8lnx)=5944x3+6720x3lnx



d6dx6(x8lnx)=5944ddx(x3)+[6720x3ddx(lnx)+(lnx)6720ddx(x3)]d6dx6(x8lnx)=5944(3x2)+6720x31x+lnx6720(3x2)d6dx6(x8lnx)=17832x2+6720x2+20160x2lnxd6dx6(x8lnx)=24552x2+20160x2lnx



d7dx7(x8lnx)=24552ddx(x2)+[20160x2ddx(lnx)+(lnx)20160ddx(x2)]d7dx7(x8lnx)=24552(2x)+20160x21x+lnx20160(2x)d7dx7(x8lnx)=49104x+20160x+40320xlnxd7dx7(x8lnx)=69264x+40320xlnx



d8dx8(x8lnx)=69264ddx(x)+[40320xddx(lnx)+(lnx)40320ddx(x)]d8dx8(x8lnx)=69264+40320x1x+40320lnxd8dx8(x8lnx)=69264+40320+40320lnxd8dx8(x8lnx)=109584+40320lnx



d9dx9(x8lnx)=ddx(109584)+40320ddx(lnx)d9dx9(x8lnx)=0+403201xd9dx9(x8lnx)=40320x

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