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If $x$ and $y$ are connected parametrically by the given equations, without eliminating the parameter. Find $\frac{d y}{d x}$
$x=a(\cos \theta+\theta \sin \theta), y=a(\sin \theta-\theta \cos \theta)$
$x=a(\cos \theta+\theta \sin \theta), y=a(\sin \theta-\theta \cos \theta)$
Solution:
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Verified Answer
$\frac{d x}{d \theta}=a[-\sin \theta+\theta \cdot \cos \theta+\sin \theta]=a \theta \cos \theta$
$\frac{d y}{d \theta}=a \theta \sin \theta \Rightarrow \frac{d y}{d x}=\frac{a \theta \sin \theta}{a \theta \cos \theta}=\tan \theta$
$\frac{d y}{d \theta}=a \theta \sin \theta \Rightarrow \frac{d y}{d x}=\frac{a \theta \sin \theta}{a \theta \cos \theta}=\tan \theta$
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