Question

If $y=x\sin y$, prove that $\frac{dy}{dx}=\frac{\sin y}{\left(1-x\cos y\right)}$

Answer 1

$\mathrm{We}\mathrm{have},y=x\sin y$
Differentiating with respect to x, we get,
$$\begin{gathered} \frac{\mathit{d}y}{\mathit{d}x}=\frac{d}{dx}\left(x\sin y\right) \\ \Rightarrow \frac{\mathit{d}y}{\mathit{d}x}=x\frac{d}{dx}\left(\sin y\right)+\sin y\frac{d}{dx}\left(x\right) \\ \Rightarrow \frac{\mathit{d}y}{\mathit{d}x}=x\cos y\frac{dy}{dx}+\sin y \\ \Rightarrow \frac{\mathit{d}y}{\mathit{d}x}\left(1-x\cos y\right)=\sin y \\ \Rightarrow \frac{\mathit{d}y}{\mathit{d}x}=\frac{\sin y}{1-x\cos y} \end{gathered}$$