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Mathematics
Alex Morrison
20 September, 00:15
Prove that dy/dx=1/x (y=lnx)
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Tristan Hammond
20 September, 04:10
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I would use the limit definition of a derivative.
where h is a very small change in x
y' = [ln (x+h) - ln (x) ]/h
Means the same as the:
Limit: (ln (x+h) - ln (x)) / h
as h - - > 0
Using the properties of logarithms, combine them into the logarithm function.
Limit: ln ((x+h) / x) / h
ln (1+h/x) * (1/h)
change of variable
h/x = n; h = nx
as h--> 0 so does n-->0
So now we have ...
Limit: (1/nx) * ln (1+n)
as n--> 0
now the definition of Euler's number "e"
Limit: (1+n) ^ (1/n) as n-->0
So we change the limit to
Limit: (1/x) * ln ((1+n) ^ (1/n))
as n-->0
"limit of a log is the log of a limit"
(1/x) * ln (Limit: (1+n) ^ (1/n) as n-->0)
= (1/x) * ln (e)
= (1/x) * (1)
= 1/x
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