Find the derivatives of the functions in Exercises 1–42.


𝓻 = ( sin θ )²
( cos θ - 1 )

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1. Identify the function given: \( r = (\sin \theta)^2 (\cos \theta - 1) \). This is a product of two functions, \( u(\theta) = (\sin \theta)^2 \) and \( v(\theta) = (\cos \theta - 1) \).
2. Apply the product rule for derivatives, which states that if \( r = u \cdot v \), then \( \frac{dr}{d\theta} = u'v + uv' \).
3. Find the derivative of \( u(\theta) = (\sin \theta)^2 \). Use the chain rule: \( u'(\theta) = 2\sin \theta \cdot \cos \theta \).
4. Find the derivative of \( v(\theta) = \cos \theta - 1 \). The derivative is \( v'(\theta) = -\sin \theta \).
5. Substitute \( u'(\theta) \), \( v(\theta) \), \( u(\theta) \), and \( v'(\theta) \) into the product rule formula: \( \frac{dr}{d\theta} = (2\sin \theta \cdot \cos \theta)(\cos \theta - 1) + ((\sin \theta)^2)(-\sin \theta) \). Simplify the expression to find the derivative.