Question

根号设y=(x+1)^2(x+2)^3/√x+3(x+4),求y'

由对数性质可知2ln(x+1)+3ln(x+2) -1/2*ln(x+3-ln(x+4)
将上式两端分别关于x求导，可得y'/y= 2/(x+1)+3/(x+2) -1/[2(x+3)- 1/(x+4)
故y'= (x+1)^2(x+2)^3 / √x+3(x+4) * { 2/(x+1)+3/(x+2) -1/[2(x+3)]- 1/(x+4)
上式结果怎么求出来的 不懂 麻烦写过程

Answer 1

y = (x+1)^2 (x+2)^3/[(x+4)√(x+3)]
lny = 2ln(x+1) + 3ln(x+2) - ln(x+4) - (1/2)ln(x+3)
y'/y = 2/(x+1) + 3/(x+2) - 1/(x+4) - (1/2)/(x+3)
y' = y[2/(x+1) + 3/(x+2) - 1/(x+4) - (1/2)/(x+3)]
将 y = (x+1)^2 (x+2)^3/[(x+4)√(x+3)] 代入上式，得
y' = [(x+1)^2 (x+2)^3][2/(x+1)+3/(x+2)-1/(x+4)-(1/2)/(x+3)]/[(x+4)√(x+3)]

追问

(x+1)^2(x+2)^3 / √x+3(x+4) 怎么求出来

追答

见解答补充