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

Differentiate both sides of \[ e^{y} \cos x = 4 + \sin(xy) \] with respect to \( x \). 1. Differentiate the left-hand side using the product rule: \[ \frac{d}{dx} \left(e^{y} \cos x\right) = e^{y} \frac{dy}{dx} \cos x + e^{y}(-\sin x) = e^{y} \cos x\,\frac{dy}{dx} - e^{y}\sin x. \] 2. Differentiate the right-hand side. The derivative of \( 4 \) is 0 and for \( \sin(xy) \) we use the chain rule: \[ \frac{d}{dx} \sin(xy) = \cos(xy) \frac{d}{dx}(xy). \] Since \[ \frac{d}{dx}(xy) = y + x\frac{dy}{dx}, \] we have: \[ \frac{d}{dx} \sin(xy) = \cos(xy)\left(y + x\frac{dy}{dx}\right). \] 3. Equate the derivatives: \[ e^{y} \cos x\,\frac{dy}{dx} - e^{y}\sin x = \cos(xy)\left(y + x\frac{dy}{dx}\right). \] 4. Collect all terms involving \(\frac{dy}{dx}\): \[ e^{y} \cos x\,\frac{dy}{dx} - \cos(xy)x\frac{dy}{dx} = e^{y}\sin x + \cos(xy)y. \] 5. Factor out \(\frac{dy}{dx}\): \[ \frac{dy}{dx}\left(e^{y}\cos x - x\cos(xy)\right) = e^{y}\sin x + \cos(xy)y. \] 6. Solve for \(\frac{dy}{dx}\): \[ \frac{dy}{dx} = \frac{e^{y}\sin x + \cos(xy)y}{e^{y}\cos x - x\cos(xy)}. \] Thus, the derivative is: \[ \frac{dy}{dx} = \frac{e^{y}\sin x + \cos(xy)y}{e^{y}\cos x - x\cos(xy)}. \]