Use implicit differentiation to find \( \frac{d y}{d x} \) \[ x \sin (3 x+2 y)=y \cos x \]

Find the derivative with respect to \( x \) for \( x\sin(3x+2y)=y\cos(x) \). Find the first derivative by following steps: - step0: Find the derivative with respect to \(x\): \(x\sin\left(3x+2y\right)=y\cos\left(x\right)\) - step1: Take the derivative: \(\frac{d}{dx}\left(x\sin\left(3x+2y\right)\right)=\frac{d}{dx}\left(y\cos\left(x\right)\right)\) - step2: Calculate the derivative: \(\sin\left(3x+2y\right)+x\cos\left(3x+2y\right)\left(3+2\frac{dy}{dx}\right)=\frac{d}{dx}\left(y\cos\left(x\right)\right)\) - step3: Calculate the derivative: \(\sin\left(3x+2y\right)+x\cos\left(3x+2y\right)\left(3+2\frac{dy}{dx}\right)=\frac{dy}{dx}\times \cos\left(x\right)-y\sin\left(x\right)\) - step4: Rewrite the expression: \(\sin\left(3x+2y\right)+x\cos\left(3x+2y\right)\left(3+2\frac{dy}{dx}\right)=\cos\left(x\right)\frac{dy}{dx}-y\sin\left(x\right)\) - step5: Move the expression to the left side: \(\sin\left(3x+2y\right)+x\cos\left(3x+2y\right)\left(3+2\frac{dy}{dx}\right)-\left(\cos\left(x\right)\frac{dy}{dx}-y\sin\left(x\right)\right)=0\) - step6: Remove the parentheses: \(\sin\left(3x+2y\right)+x\cos\left(3x+2y\right)\left(3+2\frac{dy}{dx}\right)-\cos\left(x\right)\frac{dy}{dx}+y\sin\left(x\right)=0\) - step7: Calculate: \(\sin\left(3x+2y\right)+3x\cos\left(3x+2y\right)+2x\cos\left(3x+2y\right)\frac{dy}{dx}-\cos\left(x\right)\frac{dy}{dx}+y\sin\left(x\right)=0\) - step8: Collect like terms: \(\sin\left(3x+2y\right)+3x\cos\left(3x+2y\right)+y\sin\left(x\right)+\left(2x\cos\left(3x+2y\right)-\cos\left(x\right)\right)\frac{dy}{dx}=0\) - step9: Move the constant to the right side: \(\left(2x\cos\left(3x+2y\right)-\cos\left(x\right)\right)\frac{dy}{dx}=0-\left(\sin\left(3x+2y\right)+3x\cos\left(3x+2y\right)+y\sin\left(x\right)\right)\) - step10: Subtract the terms: \(\left(2x\cos\left(3x+2y\right)-\cos\left(x\right)\right)\frac{dy}{dx}=-\sin\left(3x+2y\right)-3x\cos\left(3x+2y\right)-y\sin\left(x\right)\) - step11: Divide both sides: \(\frac{\left(2x\cos\left(3x+2y\right)-\cos\left(x\right)\right)\frac{dy}{dx}}{2x\cos\left(3x+2y\right)-\cos\left(x\right)}=\frac{-\sin\left(3x+2y\right)-3x\cos\left(3x+2y\right)-y\sin\left(x\right)}{2x\cos\left(3x+2y\right)-\cos\left(x\right)}\) - step12: Divide the numbers: \(\frac{dy}{dx}=-\frac{\sin\left(3x+2y\right)+3x\cos\left(3x+2y\right)+y\sin\left(x\right)}{2x\cos\left(3x+2y\right)-\cos\left(x\right)}\) The derivative of \( \frac{dy}{dx} \) with respect to \( x \) using implicit differentiation is given by: \[ \frac{dy}{dx} = -\frac{\sin(3x+2y)+3x\cos(3x+2y)+y\sin(x)}{2x\cos(3x+2y)-\cos(x)} \]