Question

高数，求导问题，过程

【1】f[x]=x[x+1][x+2].....[x+100]
[2]f[x]=a0 x^n+a1 x^[n-1]+..........a[n-1]x+ an

Answer 1

（1）f(x)=x（x＋1）（x＋2）（x＋3）．．．．．．（x＋100）
lnf(x)=lnx+ln(x+1)+ln(x+2)+...+ln(x+100)
[lnf(x)]'=f'(x)/f(x)=1/(x)+1/(x+1)+1/(x+2)+....+1/(x+100)
所以，
f'(x)=f(x)×[1/(x)+1/(x+1)+1/(x+2)+...+1/(x+100)]

（2）f(x)=a0 x^n+a1 x^[n-1]+..........a[n-1]x+ an
f'(x)=n×a0×x^(n-1)＋(n-1)×a1×x^(n-2)＋……＋a(n-1)

Answer 2

两边对x求导:
y' + [(1/2)/√(x +y)](x + y)'] = 1
y' + [(1/2)/√(x +y)](1 + y')] = 1