What is the derivative of #log_e(x)#?

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Answer 1 · 2018-04-04T04:36:05

#1/x#

#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)))#

#"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#