Question

If $\sin \left(xy\right)+\frac{x}{y}=x^2-y$, then find $\frac{dy}{dx}$

Answer 1

$\sin xy+\frac{x}{y}=x^2-y$

$\Rightarrow \frac{d}{dx}\left(\sin xy\right)+\frac{d}{dx}\left(\frac{x}{y}\right)=\frac{d}{dx}(x^2-y)$

$\Rightarrow \cos xy(y+x\frac{dy}{dx})+\frac{y-x\frac{dy}{dx}}{y^2}=2x-\frac{dy}{dx}$

$\Rightarrow y\cos xy+x\cos xy\frac{dy}{dx}+\frac{1}{y}-\frac{x}{y^2}\frac{dy}{dx}=2x-\frac{dy}{dx}$

$\Rightarrow \frac{dy}{dx}(1+x\cos xy-\frac{x}{y^2})=2x-y\cos xy-\frac{1}{y}$

$\frac{dy}{dx}=\frac{2x-y\cos xy-\frac{1}{y}}{x\cos xy-\frac{x}{y^2}+1}$