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Students in Calculus I were given the following problem: ``Find the derivative of $y = \sec^2{(1+3x)}$.'' Four student solutions are below. All four are incorrect. Write a correct solution and decide on a rubric of partial credit if the problm is worth 5 points. For each incorrect solution below, explain the error(s) in the solution and grade the solutions based on your rubric.

\begin{enumerate}
    \item 
    
    \[ \begin{split}
        y &= (\sec(u))^2 \qquad u = 1+3x \\
        y' &= 2 (\sec(1+3x)) (\sec{x}\tan{x})\cdot 3 \\
        &= 6 \sec{x} \tan{x} \sec{(1+3x)}
    \end{split} \]
    
    \item 
    
    \[ \begin{split}
        y' &= 2 \cdot 3 \sec{(1+3x)} \tan{(1+3x)} \\
        &= 6 \sec{(1+3x)}\tan{(1+3x)}
    \end{split} \]
    
    \item 
    
    \[ \begin{split}
        y &= \sec{(1+3x)}\cdot \sec{(1+3x)} \\
        y' &= \sec \tan{(1+3x)} \cdot 3 \sec\tan{(1+3x)} \cdot 3 \\
        &= 18 \sec \tan {(1+3x)}
    \end{split} \]
    
    \item 
    \[ y' = \tan{(1+3x)}\cdot 3 \]
\end{enumerate}


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