Question

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

Answer 1

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