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PREVIOUS ANSWERS SCALC9 2.6.013.
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Find $\frac{d y}{d x}$ by implicit differentiation.
\[
\cos (x+y)=\sin (x)+\sin (y)
\]
\[
\frac{d y}{d x}=
\]
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Step 1 ：Start by taking the derivative of both sides of the equation with respect to \(x\). Remember that \(y\) is a function of \(x\), so when differentiating terms involving \(y\), the chain rule will need to be used.

Step 2 ：Differentiating the left side gives: \(-\sin(x+y)(1+\frac{dy}{dx})\).

Step 3 ：Differentiating the right side gives: \(\cos(x) + \cos(y)\frac{dy}{dx}\).

Step 4 ：Setting these equal to each other gives the equation: \(-\sin(x+y)(1+\frac{dy}{dx}) = \cos(x) + \cos(y)\frac{dy}{dx}\).

Step 5 ：Rearrange this equation to isolate \(\frac{dy}{dx}\) on one side: \(\frac{dy}{dx}(\cos(y) + \sin(x+y)) = \cos(x) + \sin(x+y)\).

Step 6 ：Finally, solve for \(\frac{dy}{dx}\) to get: \(\frac{dy}{dx} = \frac{\cos(x) + \sin(x+y)}{\cos(y) + \sin(x+y)}\).

Step 7 ：This is the derivative of \(y\) with respect to \(x\).