#### Answer

\[\frac{{dy}}{{dx}} = 5\sec x\,{\left( {\sec x + \tan x} \right)^5}\]

#### Work Step by Step

\[\begin{gathered}
y = \,\,{\left[ {\sec x + \tan x} \right]^5} \hfill \\
\hfill \\
Use\,\,the\,\,version\,\,2\,\,of\,\,the\,\,chain\,\,rule \hfill \\
\hfill \\
\,y = {u^n} \to \frac{{dy}}{{dx}} = n{u^{n - 1}}{u^,} \hfill \\
\hfill \\
then \hfill \\
\hfill \\
\frac{{dy}}{{dx}} = 5\,{\left( {\sec x + \tan x} \right)^4}\,{\left( {\sec x + \tan x} \right)^,} \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
\frac{{dy}}{{dx}} = 5\,{\left( {\sec x + \tan x} \right)^4}\,\left( {\sec x\tan x + {{\sec }^2}x} \right) \hfill \\
\hfill \\
factor \hfill \\
\hfill \\
\frac{{dy}}{{dx}} = 5\sec x\,{\left( {\sec x + \tan x} \right)^5} \hfill \\
\end{gathered} \]