https://file.aiquickdraw.com/mathbot/file/1732101606495s57vnrqp.jpg

The given parametric equations are:

\[
x = 3(t - \sin t)
\]
\[
y = 3(1 - \cos t)
\]

To find the derivative \(\frac{dy}{dx}\), we need to compute \(\frac{dy}{dt}\) and \(\frac{dx}{dt}\), and then divide \(\frac{dy}{dt}\) by \(\frac{dx}{dt}\).

1. Compute \(\frac{dx}{dt}\):
\[
\frac{dx}{dt} = 3(1 - \cos t)
\]

2. Compute \(\frac{dy}{dt}\):
\[
\frac{dy}{dt} = 3\sin t
\]

3. Find \(\frac{dy}{dx}\) using the chain rule:
\[
\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{3\sin t}{3(1 - \cos t)} = \frac{\sin t}{1 - \cos t}
\]

Thus, the derivative \(\frac{dy}{dx}\) is:
\[
\frac{dy}{dx} = \frac{\sin t}{1 - \cos t}
\]

Would you like me to provide a graph of the curve or explore further properties of this parametric representation? 

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### Related Questions:
1. How do you calculate higher-order derivatives of parametric equations?
2. What are the geometric properties of this parametric curve?
3. Can we find the arc length of this curve for a specific interval of \(t\)?
4. What is the behavior of \(\frac{dy}{dx}\) as \(t \to 0\) or \(t \to \infty\)?
5. How would you solve for the points where the curve has a horizontal or vertical tangent?

**Tip:** Always check the interval of \(t\) to understand the full behavior of parametric curves.

Learn how to calculate the derivative of the parametric equations x = 3(t - sin t) and y = 3(1 - cos t). This solution involves key calculus concepts like the chain rule and parametric differentiation. Suitable for high school students in Grades 10-12.

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