To find $\frac{dy}{dx}$, we'll first find $\frac{dx}{d\theta}$ and $\frac{dy}{d\theta}$ and then use the chain rule:

$\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}$

Given:

$x = 2(\theta + \sin\theta)$ $y = 3(1 - \cos\theta)$

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Step 1: Find $\frac{dx}{d\theta}$

$x = 2(\theta + \sin\theta)$ Differentiate with respect to $\theta$: $\frac{dx}{d\theta} = 2\left(1 + \cos\theta\right)$

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Step 2: Find $\frac{dy}{d\theta}$

$y = 3(1 - \cos\theta)$ Differentiate with respect to $\theta$: $\frac{dy}{d\theta} = 3(0 - (-\sin\theta)) = 3\sin\theta$

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Step 3: Find $\frac{dy}{dx}$

Using the chain rule: $\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}$ Substitute the results: $\frac{dy}{dx} = \frac{3\sin\theta}{2(1 + \cos\theta)}$

Thus, the result is: $\frac{dy}{dx} = \frac{3\sin\theta}{2(1 + \cos\theta)}$

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Do you want me to expand this further or solve for specific values of $\theta$? Let me know!

5 Related Questions:

1. What is the geometrical interpretation of $\frac{dy}{dx}$ in this context?
2. How does $\frac{dy}{dx}$ behave as $\theta \to 0$?
3. What are the critical points of this parametric curve?
4. Can this parametric equation be converted into a Cartesian equation?
5. What happens to $\frac{dy}{dx}$ if $\theta = \pi$?

Tip:

Always check whether $\frac{dx}{d\theta}$ is non-zero before using the chain rule to avoid division by zero