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# If $x$ and $y$ are connected parametrically by the equations given in $x = a ( \cos \theta + \theta \sin \theta), y = a ( \sin \theta - \theta \cos \: \theta)$ without eliminating the parameter, Find $\frac{\large dy}{\large dx}$.

Toolbox:
• By chain rule we have $\large\frac{dy}{dx}=\frac{dy}{d\theta}$$\times\large\frac{d\theta}{dx} Step 1: x=a(\cos\theta+\theta\sin\theta) Differentiating with respect to \theta \large\frac{dx}{d\theta}$$=a(-\sin\theta+\sin\theta+\theta\cos\theta)$
$\quad\;\;=a\theta\cos\theta$
Step 2:
$y=a(\sin\theta-\theta\cos\theta)$
Differentiating with respect to $\theta$
$\large\frac{dy}{d\theta}$$=\cos\theta-1.\cos\theta-\theta.(-\sin\theta) \quad\;\;=a(\theta\sin\theta) Step 3: \large\frac{dy}{dx}=\frac{dy}{d\theta}$$\times\large\frac{d\theta}{dx}$
$\quad\;\;=a.\theta\sin\theta\large\frac{1}{a\theta\cos\theta}$
$\quad\;\;=\large\frac{\sin\theta}{\cos\theta}$
$\quad\;\;=\tan\theta$