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Derivative ln(x)
(Math | Calculus | Derivatives | Table Of | ln)
(d/dx) ln(x) = 1/x



Proof of (d-dx) ln(x) : by definition of e

Given: Definition of Derivative; Definition of e.
Solve:

(d-dx) ln(x) = lim(d->0) [ ln(x+d) - ln(x) ] / d = lim ln((x+d)/x) / d
= lim (1/d) ln(1 + d/x) = lim [ ln (1 + d/x)^(1/d) ].

Set u=d/x and substitute:

lim(u->0) [ ln (1 + u)^(1/(ux)) ] = 1/x ln [ lim(u->0) (1 + u)^(1/u) ]
= 1/x ln (e) (Definition of e)
= 1/x.

  
 
  

 
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