Question

If $3\sin \left(xy\right)+4\cos \left(xy\right)=5$, then $\frac{dy}{dx}=$

• A. $\frac{3\sin \left(xy\right)+4\cos \left(xy\right)}{3\sin \left(xy\right)-4\cos \left(xy\right)}$
• B. $\frac{y}{x}$
• C. $-\frac{y}{x}$
• D. $noneofthese$

Answer 1

The correct option is C $-\frac{y}{x}$

$3\frac{d}{dx}\left(\sin \left(xy\right)\right)+4\frac{d}{dx}\left(\cos \left(xy\right)\right)=\frac{d}{dx}\left(5\right)$

$\Rightarrow 3\cos \left(xy\right)\frac{d}{dx}\left(xy\right)-4\sin \left(xy\right)\frac{d}{dx}\left(xy\right)=0$

$\Rightarrow (3\cos \left(xy\right)-4\sin \left(xy\right))\frac{d}{dx}\left(xy\right)=0$

$\Rightarrow (3\cos \left(xy\right)-4\sin \left(xy\right))(x\frac{dy}{dx}+y)=0$

$\Rightarrow 3\cos \left(xy\right)-4\sin \left(xy\right)=0$ and $x\frac{dy}{dx}+y=0$

$\Rightarrow 3\cos \left(xy\right)=4\sin \left(xy\right)$ and $x\frac{dy}{dx}=-y$

$\Rightarrow \tan xy=\frac{3}{4}$ and $\frac{dy}{dx}=\frac{-y}{x}$

$\therefore \frac{dy}{dx}=\frac{-y}{x}$