let y=x^(1/x)
dy/dx=(1/x)x^(1/x-1)=x^(1/x-2)
but if y=x^(1/x)
lny=(1/x)lnx
(1/y)dy/dx=(1/x)(1/x)+(lnx)(-1/x^2)
dy/dx=x^(1/x-2)(1/lnx)
What's wrong?I can't use power rule in this case? explain?
Problem about differentiation
2011-07-23 9:46 pm
回答 (2)
2011-07-23 9:55 pm
✔ 最佳答案
The power rule can only be used for constant indices. Say, for example, y=x^3 or y=x^2, then you can use this rule. 
2011-07-24 5:13 pm
Alternative derivations include:
x^(1/x)=e^(ln x / x)
Differentiating both sides w.r.t. x yields:
d/dx x^(1/x)=e^(ln x / x) * (1-ln x) / x^2
x^(1/x)=e^(ln x / x)
Differentiating both sides w.r.t. x yields:
d/dx x^(1/x)=e^(ln x / x) * (1-ln x) / x^2
收錄日期: 2021-04-22 00:54:11
原文連結 [永久失效]:
https://hk.answers.yahoo.com/question/index?qid=20110723000051KK00464