Let 3 (xy)+4 (xy)=5, then dy dx = - Vidyadip

MCQ

Let $3\sin(\text{xy})+4\cos(\text{xy})=5,$ then $\frac{\text{dy}}{\text{dx}}=$

✓
$-\frac{\text{y}}{\text{x}}$
B
$\frac{3\sin(\text{xy})+4\cos(\text{xy})}{3\cos(\text{xy})-4\sin(\text{xy})}$
C
$\frac{3\cos(\text{xy})+4\sin(\text{xy})}{4\cos(\text{xy})-3\sin(\text{xy})}$
D
None of these.

✓

Answer

Correct option: A.

$-\frac{\text{y}}{\text{x}}$

We have, $3\sin(\text{xy})+4\cos(\text{xy})=5$
$\Rightarrow3\cos(\text{xy})\Big[\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}\Big]-4\sin(\text{xy})\Big[\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}\Big]=0$
$\Rightarrow\Big[\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}\Big]\big[3\cos(\text{xy})-4\sin(\text{xy})\big]=0$
$\Rightarrow\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=0$
$\Rightarrow\text{x}\frac{\text{dy}}{\text{dx}}=-\text{y}$
$\therefore\frac{\text{dy}}{\text{dx}}=-\frac{\text{y}}{\text{x}}$

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