Question Detail

Differentiate each term with respect to $x$, using the product rule and chain rule where necessary:$\frac{d}{dx}(x^2 \sin y) = (2x)(\sin y) + (x^2)(\cos y \frac{dy}{dx})$$\frac{d}{dx}(-e^{2y} \cos x) = -[(2e^{2y} \frac{dy}{dx})(\cos x) + (e^{2y})(-\sin x)] = -2e^{2y}\cos x \frac{dy}{dx} + e^{2y}\sin x$$\frac{d}{dx}(3x^2 + 2x - 5) = 6x + 2$Combine these to form the differentiated equation:$2x\sin y + x^2\cos y \frac{dy}{dx} - 2e^{2y}\cos x \frac{dy}{dx} + e^{2y}\sin x = 6x + 2$Group terms with $\frac{dy}{dx}$ and move other terms to the right side:$\frac{dy}{dx}(x^2\cos y - 2e^{2y}\cos x) = 6x + 2 - 2x\sin y - e^{2y}\sin x$Finally, solve for $\frac{dy}{dx}$:$\frac{dy}{dx} = \frac{6x + 2 - 2x\sin y - e^{2y}\sin x}{x^2\cos y - 2e^{2y}\cos x}$Rearranging the numerator for a better match with option C:$\frac{dy}{dx} = \frac{6x - 2x\sin y - e^{2y}\sin x + 2}{x^2\cos y - 2e^{2y}\cos x}$.