#### Answer

\[y'' = 2{\sec ^2}x\tan x\]

#### Work Step by Step

\[\begin{gathered}
y = \tan x \hfill \\
\hfill \\
find\,\,{y^{\,,}} \hfill \\
\hfill \\
y' = {\sec ^2}x \hfill \\
\hfill \\
differentiate\,\,to\,\,find\,\,{y^,}^, \hfill \\
use\,\,the\,\,chain\,\,\,rule \hfill \\
\hfill \\
y'' = 2\,\left( {\sec x} \right)\,{\left( {\sec x} \right)^\prime } \hfill \\
\hfill \\
so \hfill \\
\hfill \\
y'' = 2\,\left( {\sec x} \right)\,\left( {\sec x\tan x} \right) \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
y'' = 2{\sec ^2}x\tan x \hfill \\
\hfill \\
\end{gathered} \]