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Find $y^{\prime }$ by implicit differentiation. Match the equations defining $y$ implicitly with the letters labeling the expressions for $y^{\prime }$.
1. $7x\cos y+2\sin 2y=4\sin y$
2. $7x\cos y+2\cos 2y=4\sin y$
3. $7x\sin y+2\sin 2y=4\cos y$
4. $7x\sin y+2\cos 2y=4\cos y$
A. $y^{\prime }=\frac{7\cos y}{7x\sin y-4\cos 2y+4\cos y}$
B. $y^{\prime }=\frac{7\sin y}{-7x\cos y-4\cos 2y-4\sin y}$
C. $y^{\prime }=\frac{7\sin y}{4\sin 2y-7x\cos y-4\sin y}$
D. $y^{\prime }=\frac{7\cos y}{7x\sin y+4\sin 2y+4\cos y}$
Note: You can earn partial credit on this problem.

Step 1 ：Differentiate both sides of the first equation $7x\cos y+2\sin 2y=4\sin y$ with respect to $x$ using the chain rule and the product rule.

Step 2 ：The derivative of $7x\cos y$ with respect to $x$ is $7\cos y-7x\sin y\cdot y^{\prime }$.

Step 3 ：The derivative of $2\sin 2y$ with respect to $x$ is $4\cos 2y\cdot y^{\prime }$.

Step 4 ：The derivative of $4\sin y$ with respect to $x$ is $4\cos y\cdot y^{\prime }$.

Step 5 ：Setting these equal gives us $7\cos y-7x\sin y\cdot y^{\prime }+4\cos 2y\cdot y^{\prime }=4\cos y\cdot y^{\prime }$.

Step 6 ：Solving for $y^{\prime }$ gives us $y^{\prime }=\frac{7\cos y}{7x\sin y+4\cos 2y-4\cos y}$, which matches with option D.

Step 7 ：Differentiate both sides of the third equation $7x\sin y+2\sin 2y=4\cos y$ with respect to $x$ using the chain rule and the product rule.

Step 8 ：The derivative of $7x\sin y$ with respect to $x$ is $7\sin y+7x\cos y\cdot y^{\prime }$.

Step 9 ：The derivative of $2\sin 2y$ with respect to $x$ is $4\cos 2y\cdot y^{\prime }$.

Step 10 ：The derivative of $4\cos y$ with respect to $x$ is $-4\sin y\cdot y^{\prime }$.

Step 11 ：Setting these equal gives us $7\sin y+7x\cos y\cdot y^{\prime }+4\cos 2y\cdot y^{\prime }=-4\sin y\cdot y^{\prime }$.

Step 12 ：Solving for $y^{\prime }$ gives us $y^{\prime }=\frac{7\sin y}{-7x\cos y-4\cos 2y+4\sin y}$, which matches with option B.

Step 13 ：So, the first equation matches with option D and the third equation matches with option B.