Find \( d y / d x \) by implicit differentiation. \[ e^{y} \cos (x)=4+\sin (x y) \] \( \frac{d y}{d x}=\square \)

To find \( \frac{d y}{d x} \) using implicit differentiation for the equation \( e^{y} \cos (x) = 4 + \sin (x y) \), we'll differentiate both sides with respect to \( x \). Differentiating the left side: \[ \frac{d}{dx}(e^{y} \cos(x)) = e^{y} \frac{dy}{dx} \cos(x) - e^{y} \sin(x) \] Differentiating the right side: \[ \frac{d}{dx}(4 + \sin(xy)) = \cos(xy)(y + x \frac{dy}{dx}) \] Setting the derivatives equal to each other gives: \[ e^{y} \frac{dy}{dx} \cos(x) - e^{y} \sin(x) = \cos(xy)(y + x \frac{dy}{dx}) \] Now, rearranging the terms to isolate \( \frac{dy}{dx} \): \[ e^{y} \frac{dy}{dx} \cos(x) - x \cos(xy) \frac{dy}{dx} = \cos(xy) y + e^{y} \sin(x) \] Factoring out \( \frac{dy}{dx} \): \[ \frac{dy}{dx}(e^{y} \cos(x) - x \cos(xy)) = \cos(xy) y + e^{y} \sin(x) \] Now, solving for \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \frac{\cos(xy) y + e^{y} \sin(x)}{e^{y} \cos(x) - x \cos(xy)} \] So, the final answer is: \[ \frac{d y}{d x} = \frac{\cos(xy) y + e^{y} \sin(x)}{e^{y} \cos(x) - x \cos(xy)} \]