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 \(x\).