11. (6 points) Consider the polar curve $r=1-\sin \theta$
(a) (3 points) Find $\frac{d x}{d \theta}$
(b) (3 points) $\frac{d y}{d \theta}$

Step 1 ：We are given the polar curve \(r=1-\sin \theta\).

Step 2 ：We know that in polar coordinates, \(x = r\cos\theta\) and \(y = r\sin\theta\).

Step 3 ：We need to find the derivatives of these expressions with respect to \(\theta\).

Step 4 ：For \(\frac{dx}{d\theta}\), we will use the product rule for differentiation which states that the derivative of a product of two functions is the derivative of the first times the second plus the first times the derivative of the second.

Step 5 ：For \(\frac{dy}{d\theta}\), we will also use the product rule for differentiation.

Step 6 ：Applying the product rule, we find that \(\frac{dx}{d\theta} = -(1 - \sin\theta)\sin\theta - \cos^2\theta\) and \(\frac{dy}{d\theta} = (1 - \sin\theta)\cos\theta - \sin\theta\cos\theta\).

Step 7 ：\(\boxed{\frac{dx}{d\theta} = -(1 - \sin\theta)\sin\theta - \cos^2\theta}\) and \(\boxed{\frac{dy}{d\theta} = (1 - \sin\theta)\cos\theta - \sin\theta\cos\theta}\) are the final answers.