Find d y d x , if x y+y x=0 - Vidyadip

Question

Find $\frac{d y}{d x}$, if $x \sin y+y \sin x=0$

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Answer

x sin y + y sin x = 0
Differentiate w.r.t. x both side
$\begin{aligned} & {\left[x \cos y \frac{d y}{d x}+\sin y\right]+\left[y \cos x+\sin x \frac{d y}{d x}\right]=0} \\ & \therefore \sin y+y \cos x=\frac{d y}{d x}(-\sin x-x \cos y) \\ & \therefore \frac{d y}{d x}=-\left(\frac{\sin y+y \cos x}{\sin x+x \cos y}\right)\end{aligned}$

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