Question

what is the derivative lnx

Answer 1

your answer is 1/x
thanks for asking
.........

Answer 2

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)
= 1/x......