Maths M1 proof question
2013-05-25 1:57 am
Is it possible to prove that (ln x)' = 1/x,without using inverse function rule (dx/dy = 1/(dy/dx)), nor the firstprinciples, nor differentiating both sides with respect to x? Thank you.
回答 (2)
2013-05-25 7:58 pm
✔ 最佳答案
dx/dx=1d(ln e^x)/dx=1
d(ln e^x)/d(e^x) * d(e^x)/dx = 1
d(ln e^x)/d(e^x) * e^x = 1
d(ln e^x)/d(e^x) = 1/e^x
Let e^x=y
d(ln y)/dy = 1/y
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2013-05-25 2:32 am
Geometrically
2013-05-25 10:21:29 補充：
using the curse lnx and draw tangent lines and finding out the slopes of the tangent and then using the slopes to find out the equation joining them by using Lagrange's interpolation formula. When the number of slope you increases, the equation joining the slopes will thus tend to 1/x.
2013-05-25 10:21:29 補充：
using the curse lnx and draw tangent lines and finding out the slopes of the tangent and then using the slopes to find out the equation joining them by using Lagrange's interpolation formula. When the number of slope you increases, the equation joining the slopes will thus tend to 1/x.
參考: Simon YAU, Simon YAU
收錄日期: 2021-04-13 19:29:26
原文連結 [永久失效]:
https://hk.answers.yahoo.com/question/index?qid=20130524000051KK00191