Question

一道大学高数求导题设y=x²ln(x+1)求y的三阶导和五十阶导

Answer 1

y=x²ln(x+1)

y’=2xln(x+1)+x²/(x+1)=2xln(x+1)+(x+1)+1/(x+1)-2

y’’=2ln(x+1)+2x/(x+1)+1-1/(x+1)^2=2ln(x+1)-2/(x+1)-1/(x+1)^2+3

y’’’=2/(x+1)+2/(x+1)^2+2(x+1)^3

……

y(n)=2(n-2)!(-1)^(n+1)[1/(x+1)^(n-2)+1/(x+1)^(n-1)+1/(x+1)^n-1

(n≥3)

y(50)=-2*48![1/(x+1)^48+1/(x+1)^49+1/(x+1)^50]