# Choose the correct alternative in the following:

Question:

Choose the correct alternative in the following:

If $3 \sin (x y)+4 \cos (x y)=5$, then $\frac{d y}{d x}=$

A. $-\frac{\mathrm{y}}{\mathrm{x}}$

B. $\frac{3 \sin (x y)+4 \cos (x y)}{3 \cos (x y)-4 \sin (x y)}$

C. $\frac{3 \cos (\mathrm{xy})+4 \sin (\mathrm{xy})}{4 \cos (\mathrm{xy})-3 \sin (\mathrm{xy})}$

D. none of these

Solution:

$3 \sin (x y)+4 \cos (x y)=5$

Differentiating w.r.t $x$ we get,

$\Rightarrow 3\left[\cos (\mathrm{xy}) \cdot\left(1 \cdot \mathrm{y}+\mathrm{x} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}\right)\right]+4\left[-\sin (\mathrm{xy}) \cdot\left(1 \cdot \mathrm{y}+\mathrm{x} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}\right)\right]=0$

(Using Chain Rule)

$\Rightarrow\left[3 y \cos (x y)+3 x \cos (x y) \cdot \frac{d y}{d x}\right]+\left[-4 y \sin (x y)-4 x \sin (x y) \cdot \frac{d y}{d x}\right]=0$

$\Rightarrow \frac{d y}{d x}[3 x \cos (x y)-4 x \sin (x y)]=4 y \sin (x y)-3 y \cos (x y)$

$\Rightarrow \frac{d y}{d x}=-\frac{y[-4 \sin (x y)+3 \cos (x y)]}{x[3 \cos (x y)-4 \sin (x y)]}=-\frac{y}{x}$

$\Rightarrow \frac{d y}{d x}=-\frac{y}{x}=(A)$