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If $x siny = 3 siny + 4 cosy$ then $\frac{dy}{dx} =$

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a

\frac{\sin^{2} y}{4}

b

\frac{\cos^{2} y}{4}

c

\frac{- \sin^{2} y}{4}

d

\frac{- \cos^{2} y}{4}

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detailed solution

Correct option is C

The given equation is
$x \sin y = 3 \sin y + 4 \cos y \Rightarrow (x - 3) \sin y = 4 \cos y$
$\Rightarrow x - 3 = 4 c o t y \Rightarrow x = 4 c o t y + 3 d i f f e r e n t i a t e w . r . t y$
$\Rightarrow \frac{d x}{d y} = - 4 \cos e c^{2} y = \frac{- 4}{\sin^{2} y} \Rightarrow \frac{d y}{d x} = \frac{- \sin^{2} y}{4}$

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