Question

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

Answer 1

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`