#### Answer

\[y = \frac{3}{2}{x^{1/2}} + \frac{1}{2}{x^{ - 1/2}} + 2\]

#### Work Step by Step

\[\begin{gathered}
y = \,\left( {x + 1} \right)\,\left( {\sqrt x + 2} \right) \hfill \\
\sqrt x \,\,can\,\,be\,\,\,written\,\,as\,\,{x^{1/2}} \hfill \\
y = \,\left( {x + 1} \right)\,\left( {{x^{1/2}} + 2} \right) \hfill \\
Use\,\,the\,\,product\,\,rule\,\,to\,\,find\,\,{y^,} \hfill \\
{y^,} = \,\left( {x + 1} \right)\,{\left( {{x^{1/2}} + 2} \right)^,} + \,\left( {{x^{1/2}} + 2} \right)\,{\left( {x + 1} \right)^,} \hfill \\
Then \hfill \\
{y^,} = \,\left( {x + 1} \right)\,\left( {\frac{1}{2}{x^{ - 1/2}}} \right) + \,\left( {{x^{1/2}} + 2} \right)\,\left( 1 \right) \hfill \\
Simplify\,\,by\,\,multiplying\,\,and\,\,combining\,\,terms \hfill \\
{y^,} = \frac{1}{2}{x^{1/2}} + \frac{1}{2}{x^{ - 1/2}} + {x^{1/2}} + 2 \hfill \\
y = \frac{3}{2}{x^{1/2}} + \frac{1}{2}{x^{ - 1/2}} + 2 \hfill \\
\end{gathered} \]