1. The problem is to find the derivative $S'(t)$ given as $|\sec(\frac{\pi t}{12})|$. 2. The function involves the absolute value of the secant function, which is $|\sec(x)|$ where $x=\frac{\pi t}{12}$. 3. To differentiate $S'(t) = |\sec(\frac{\pi t}{12})|$, we use the chain rule and the derivative of the secant function. 4. Recall that $\frac{d}{dx} \sec(x) = \sec(x) \tan(x)$. 5. The derivative of the absolute value function $|u|$ with respect to $t$ is $\frac{d}{dt} |u| = \frac{u}{|u|} \frac{du}{dt}$ for $u \neq 0$. 6. Let $u = \sec(\frac{\pi t}{12})$, then $\frac{du}{dt} = \sec(\frac{\pi t}{12}) \tan(\frac{\pi t}{12}) \cdot \frac{\pi}{12}$ by the chain rule. 7. Therefore, $$ S'(t) = \frac{u}{|u|} \cdot \frac{du}{dt} = \frac{\sec(\frac{\pi t}{12})}{|\sec(\frac{\pi t}{12})|} \cdot \sec(\frac{\pi t}{12}) \tan(\frac{\pi t}{12}) \cdot \frac{\pi}{12} = \frac{\pi}{12} \cdot \frac{\sec^2(\frac{\pi t}{12}) \tan(\frac{\pi t}{12})}{|\sec(\frac{\pi t}{12})|} $$ 8. Since $|\sec(x)| = \sqrt{\sec^2(x)}$, the expression simplifies to $$ S'(t) = \frac{\pi}{12} \cdot |\sec(\frac{\pi t}{12})| \tan(\frac{\pi t}{12}) $$ 9. This is the derivative of the given function in terms of $t$. Final answer: $$ S'(t) = \frac{\pi}{12} |\sec(\frac{\pi t}{12})| \tan(\frac{\pi t}{12}) $$