How do I us the Limit definition of derivative on #f(x)=ln(x)#?

1 Answer
Sep 17, 2014

By Limit Definition of the Derivative,
#f'(x)=1/x#.

Let us look at some details.

By Limit Definition,

#f'(x)=lim_{h to 0}{ln(x+h)-lnx}/h#

by the log property #lna-lnb=ln(a/b)#,

#=lim_{h to 0}ln({x+h}/x)/h#

by rewriting a bit further,

#=lim_{h to 0}1/hln(1+h/x)#

by the log property #rlnx=lnx^r#,

#=lim_{h to 0}ln(1+h/x)^{1/h}#

by the substitution #t=h/x# (#Leftrightarrow 1/h=1/{tx}#),

#=lim_{t to 0}ln(1+t)^{1/{t}cdot1/x}#

by the log property #lnx^r=rlnx#,

#=lim_{t to 0}1/xln(1+t)^{1/t}#

by pulling #1/x# out of the limit,

#=1/xlim_{t to 0}ln(1+t)^{1/t}#

by putting the limit inside the natural log,

#1/xln[lim_{t to 0}(1+t)^{1/t}]#

by the definition #e=lim_{t to 0}(1+t)^{1/t}#,

#=1/xlne=1/x cdot 1=1/x#