Question

1.利用对数求导法求导:y=[(x-1)x(x+1)(x+2)/(x-2)^2(x+3)]^1/3 2.求二阶导数（y`

Answer 1

1.lny=1/3[ln(x-1)+lnx+ln(x+1)+ln(x+2)-2ln(x-2)-ln(x+3)]

y'/y=1/3[1/(x-1)+1/x+1/(x+1)+1/(x+2)-2/(x-2)-1/(x+3)]

y'=1/3[1/(x-1)+1/x+1/(x+1)+1/(x+2)-2/(x-2)-1/(x+3)][(x-1)x(x+1)(x+2)/(x-2)^2(x+3)]^1/3

2.y'=10(x^2-2x+5)^9(2x-2)=20(x^2-2x+5)^9(x-1)

y"=20[(x^2-2x+5)^9]'(x-1)+20(x^2-2x+5)^9[(x-1)]'

=20[9(x^2-2x+5)^8(x-1)²]+20(x^2-2x+5)^9

=20(x²-2x+5)^8[9(x-1)²+x²-2x+5]

3.lim（x趋向0+）lnx/lnsinx =lim 1/x /(cosx/sinx)=lim sinx/(xcosx)=lim tanx/x=1

4.y'=1-e^x>=0 e^x