A) \[\frac{y[2xy-{{y}^{2}}\cos (xy)-1]}{x{{y}^{2}}\cos (xy)+{{y}^{2}}-x}\]
B) \[\frac{[2xy-{{y}^{2}}\cos (xy)-1]}{x{{y}^{2}}\cos (xy)+{{y}^{2}}-x}\]
C) \[-\frac{y[2xy-{{y}^{2}}\cos (xy)-1]}{x{{y}^{2}}\cos (xy)+{{y}^{2}}-x}\]
D) None of these
Correct Answer: A
Solution :
\[\sin (xy)+\frac{x}{y}={{x}^{2}}-y\] Differentiating both sides, \[\cos (xy)\frac{d}{dx}(xy)+x\left\{ -\frac{1}{{{y}^{2}}} \right\}\frac{dy}{dx}+\frac{1}{y}=2x-\frac{dy}{dx}\] Þ \[[x\cos (xy)-\frac{x}{{{y}^{2}}}+1]\frac{dy}{dx}=2x-\frac{1}{y}-y\cos (xy)\] Þ \[\frac{dy}{dx}=\left[ \frac{2x{{y}^{2}}-y-{{y}^{3}}\cos (xy)}{x{{y}^{2}}\cos (xy)-x+{{y}^{2}}} \right]\].
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