X(X-1)(X-2).....(X-n)求导......

计算过程为：

设y=x(x-1)(x-2).....(x-n)，则两边取对数可以得到

lny=ln[x(x-1)(x-2).....(x-n)]

=lnx+ln(x-1)+ln(x-2)+.....+ln(x-n)

对两边求导：

y'/y=1/x+1/(x-1)+1/(x-2)+.....+1/(x-n)

则

y'=y*[1/x+1/(x-1)+1/(x-2)+.....+1/(x-n)]

=(x-1)(x-2)...(x-n)+x(x-2)...(x-n)+x(x-1)(x-3)...(x-n)+...+x(x-1)(x-2).....(x-n+1)

扩展资料：

常用求导公式

1、C'=0(C为常数)

2、(Xn)'=nX(n-1) (n∈R)

3、(sinX)'=cosX

4、(cosX)'=-sinX

5、(aX)'=aXIna （ln为自然对数)

6、(logaX)'=（1/X)logae=1/(Xlna) (a>0，且a≠1)

7、(tanX)'=1/(cosX)²=(secX)²

8、(cotX)'=-1/(sinX)²=-(cscX)²

9、(secX)'=tanXsecX

10、(cscX)'=-cotXcscX

设y=x(x-1)(x-2).....(x-n)
lny=ln[x(x-1)(x-2).....(x-n)]
lny=lnx+ln(x-1)+ln(x-2)+.....+ln(x-n)两边求导
y'/y=1/x+1/(x-1)+1/(x-2)+.....+1/(x-n)
y'=y*[1/x+1/(x-1)+1/(x-2)+.....+1/(x-n)]
=(x-1)(x-2)...(x-n)
+x(x-2)...(x-n)
+x(x-1)(x-3)...(x-n)
+...
+x(x-1)(x-2).....(x-n+1)

(X-1)(X-2).....(X-n)+
X(X-2).....(X-n)+
X(X-1)(X-3).....(X-n)+
...
X(X-1)(X-2).....(X-n+1)