Question

If $x$ $sin$ $y$$=3siny$ $+$ $4cosy$, then $\frac{dy}{dx}=$

• A. $\frac{{-sin}^{2}y}{4}$
• B. $\frac{{sin}^{2}y}{4}$
• C. $\frac{{-cos}^{2}y}{4}$
• D. $\frac{{cos}^{2}y}{4}$

Answer 1

The correct option is A $\frac{{-sin}^{2}y}{4}$

$x\sin y=3\sin y+4\cos y$

$\Rightarrow (x-3)\sin y=4\cos y$

$\Rightarrow x-3=4\frac{\cos y}{\sin y}$

$\Rightarrow x-3=4\cot y$

Differentiating both sides w.r.t $x$

$\Rightarrow 1=4(-{\csc }^{2}y)\frac{dy}{dx}$

$\Rightarrow \frac{dy}{dx}=\frac{-1}{4{\csc }^{2}y}$

$\therefore \frac{dy}{dx}=\frac{-{\sin }^{2}y}{4}$