Answer
$${\bf{N}}\left( 2 \right) = - \frac{2}{{\sqrt 5 }}{\bf{i}} + \frac{1}{{\sqrt 5 }}{\bf{j}}$$
Work Step by Step
$$\eqalign{
& {\bf{r}}\left( t \right) = \ln t{\bf{i}} + \left( {t + 1} \right){\bf{j}},{\text{ }}t = 2 \cr
& {\text{Calculate }}{\bf{r}}{\text{'}}\left( t \right) \cr
& {\bf{r}}'\left( t \right) = \frac{1}{t}{\bf{i}} + {\bf{j}} \cr
& {\text{Find the unit Tangent Vector }}{\bf{T}}\left( t \right) \cr
& {\bf{T}}\left( t \right) = \frac{{{\bf{r}}'\left( t \right)}}{{\left\| {{\bf{r}}'\left( t \right)} \right\|}},{\text{ }}{\bf{r}}'\left( t \right) \ne 0 \cr
& {\bf{T}}\left( t \right) = \frac{{\frac{1}{t}{\bf{i}} + {\bf{j}}}}{{\left\| {\frac{1}{t}{\bf{i}} + {\bf{j}}} \right\|}} = \frac{{\frac{1}{t}{\bf{i}} + {\bf{j}}}}{{\sqrt {\frac{1}{{{t^2}}} + 1} }} = \frac{{\frac{1}{t}{\bf{i}} + {\bf{j}}}}{{\frac{{\sqrt {1 + {t^2}} }}{t}}} \cr
& {\bf{T}}\left( t \right) = \frac{{{\bf{i}} + t{\bf{j}}}}{{\sqrt {1 + {t^2}} }} \cr
& {\text{Find }}{\bf{T}}'\left( t \right) \cr
& {\bf{T}}'\left( t \right) = \frac{d}{{dt}}\left[ {\frac{1}{{\sqrt {1 + {t^2}} }}} \right]{\bf{i}} + \frac{d}{{dt}}\left[ {\frac{t}{{\sqrt {1 + {t^2}} }}} \right]{\bf{j}} \cr
& {\text{Making the operations by hand and replacing }}\left( {{\text{or use a CAS}}} \right) \cr
& {\bf{T}}'\left( t \right) = - \frac{t}{{{{\left( {1 + {t^2}} \right)}^{3/2}}}}{\bf{i}} + \frac{1}{{{{\left( {1 + {t^2}} \right)}^{3/2}}}}{\bf{j}} \cr
& {\text{Evaluate }}{\bf{T}}'\left( t \right){\text{ at }}t = 2 \cr
& {\bf{T}}'\left( 2 \right) = - \frac{2}{{{{\left( {1 + {2^2}} \right)}^{3/2}}}}{\bf{i}} + \frac{1}{{{{\left( {1 + {2^2}} \right)}^{3/2}}}}{\bf{j}} \cr
& {\bf{T}}'\left( 2 \right) = - \frac{2}{{{{\left( 5 \right)}^{3/2}}}}{\bf{i}} + \frac{1}{{{{\left( 5 \right)}^{3/2}}}}{\bf{j}} \cr
& \left\| {{\bf{T}}'\left( 2 \right)} \right\| = \sqrt {{{\left( { - \frac{2}{{5\sqrt 5 }}} \right)}^2} + {{\left( {\frac{1}{{5\sqrt 5 }}} \right)}^2}} = \sqrt {\frac{1}{{25}}} = \frac{1}{5} \cr
& {\text{Finding the Principal Unit Normal Vector}} \cr
& {\bf{N}}\left( t \right) = \frac{{{\bf{T}}'\left( t \right)}}{{\left\| {{\bf{T}}'\left( t \right)} \right\|}} \cr
& {\text{Evaluate }}{\bf{N}}'\left( t \right){\text{ at }}t = 2 \cr
& {\bf{N}}\left( 2 \right) = \frac{{{\bf{T}}'\left( 2 \right)}}{{\left\| {{\bf{T}}'\left( 2 \right)} \right\|}} = \frac{{ - \frac{2}{{{{\left( 5 \right)}^{3/2}}}}{\bf{i}} + \frac{1}{{{{\left( 5 \right)}^{3/2}}}}{\bf{j}}}}{{1/5}} \cr
& {\bf{N}}\left( 2 \right) = - \frac{{2\left( 5 \right)}}{{{{\left( 5 \right)}^{3/2}}}}{\bf{i}} + \frac{5}{{{{\left( 5 \right)}^{3/2}}}}{\bf{j}} \cr
& {\bf{N}}\left( 2 \right) = - \frac{2}{{\sqrt 5 }}{\bf{i}} + \frac{1}{{\sqrt 5 }}{\bf{j}} \cr} $$
