[tex]\bf 2^{\cfrac{}{}log_2\left(\frac{x^2+2x+2}{2} \right)}
\\
-----------------------------\\\\
\textit{recall the log cancellation rule of }\qquad
{{ a}}^{log_{{ a}}x}=x\\
-----------------------------\\
\\
thus
\\\\
2^{\cfrac{}{}log_2\left(\frac{x^2+2x+2}{2} \right)}\impliedby \textit{base of log and expression are the same, thus}
\\\\\\
\implies \cfrac{x^2+2x+2}{2}\implies \cfrac{x^2}{2}+\cfrac{2x}{2}+\cfrac{2}{2}\implies
\cfrac{1}{2}x^2+x+1[/tex]
and pretty sure you can take the derivative of that
