Derivative ln(x)

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Derivative ln(x)

(Math | Calculus | Derivatives | Table Of | ln)

ln(x) = 1/x

Proof of

ln(x) : by definition of e

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

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.