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Question
If \[3 \sin \left( xy \right) + 4 \cos \left( xy \right) = 5, \text { then } \frac{dy}{dx} =\] _____________ .
Options
\[- \frac{y}{x}\]
\[\frac{3 \sin \left( xy \right) + 4 \cos \left( xy \right)}{3 \cos \left( xy \right) - 4 \sin \left( xy \right)}\]
\[\frac{3 \cos \left( xy \right) + 4 \sin \left( xy \right)}{4 \cos \left( xy \right) - 3 \sin \left( xy \right)}\]
none of these
Solution
\[- \frac{y}{x}\]
\[\text { We have, } 3 \sin\left( xy \right) + 4 \cos\left( xy \right) = 5 \]
\[ \Rightarrow 3 \cos\left( xy \right)\left[ x\frac{dy}{dx} + y \right] - 4 \sin\left( xy \right)\left[ x\frac{dy}{dx} + y \right] = 0\]
\[ \Rightarrow \left[ x\frac{dy}{dx} + y \right]\left[ 3 \cos\left( xy \right) - 4 \sin\left( xy \right) \right] = 0\]
\[ \Rightarrow x\frac{dy}{dx} + y = 0\]
\[ \Rightarrow x\frac{dy}{dx} = - y\]
\[ \therefore \frac{dy}{dx} = - \frac{y}{x}\]
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