#log_e(x)# is commonly denoted as #ln(x)#, the natural log.
#=>d/(dx) ln(x) = 1/x#
If you would like a proof, we can derive it from the limit definition:
#lim_(delta x->0)(f(x+delta x)-f(x))/(delta x)#
#= lim_(delta x->0)(ln(x+delta x)-ln(x))/(delta x)#
#= lim_(delta x->0)(ln((x+delta x)/(x)))/(delta x)#
#= lim_(delta x->0)1/(delta x)ln(1+(delta x)/x)#
#= lim_(delta x->0)ln((1+(delta x)/x)^(1/(delta x)))#
#= lim_(delta x->0)ln((1+(delta x)/x)^(1/(delta x)))#
#"Let " tau equiv (delta x)/x#:
#= lim_(delta tau->0)ln((1+tau)^(1/(xtau)))#
#= lim_(delta tau->0)ln[((1+tau)^(1/(tau)))^(1/x)]#
#= ln[(e)^(1/x)]#
#= 1/x ln(e)#
#= 1/x (1)#
#= 1/x#