Answer
\[y = 2ex - e\]
Work Step by Step
\[\begin{gathered}
y = x{e^x} \hfill \\
Evaluate\,\,the\,\,function\,\,at\,\,x = 1 \hfill \\
y\,\left( 1 \right) = \,\left( 1 \right){e^{\,\left( 1 \right)}} \hfill \\
y\,\left( 1 \right) = e \hfill \\
P\,oint\,\,\,\left( {1,e} \right) \hfill \\
Find\,\,the\,\,deriva\,tive\,\,of\,\,the\,\,function \hfill \\
{y^,} = \,\,{\left[ {x{e^x}} \right]^,} \hfill \\
Use\,\,the\,\,product\,\,rule \hfill \\
{y^,} = x{e^x} + {e^x} \hfill \\
Evaluate\,\,{y^,}\,\,at\,\,x = 1 \hfill \\
m = {y^,}\,\left( 1 \right) = e + e \hfill \\
m = 2e \hfill \\
Use\,\,the\,\,point\, - slope\,\,form \hfill \\
y - {y_1} = m\,\left( {x - {x_1}} \right) \hfill \\
y - e = 2e\,\left( {x - 1} \right) \hfill \\
y - e = 2ex - 2e \hfill \\
y = 2ex - e \hfill \\
\end{gathered} \]
