Question Detail

To differentiate implicitly, apply the derivative operator $d/dx$ to both sides of the equation:$d/dx(x^2 \sin y) - d/dx(e^{2y} \cos x) = d/dx(3x^2 + 2x - 5)$Using the product rule and chain rule:$d/dx(x^2 \sin y) = 2x \sin y + x^2 \cos y \frac{dy}{dx}$$d/dx(e^{2y} \cos x) = (2e^{2y} \frac{dy}{dx}) \cos x + e^{2y} (-\sin x) = 2e^{2y} \cos x \frac{dy}{dx} - e^{2y} \sin x$$d/dx(3x^2 + 2x - 5) = 6x + 2$Substitute these back into the original equation:$(2x \sin y + x^2 \cos y \frac{dy}{dx}) - (2e^{2y} \cos x \frac{dy}{dx} - e^{2y} \sin x) = 6x + 2$$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 $\frac{dy}{dx}$:$\frac{dy}{dx} (x^2 \cos y - 2e^{2y} \cos x) = 6x + 2 - 2x \sin y - e^{2y} \sin x$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 terms matches option C: $\frac{dy}{dx} = \frac{6x - 2x \sin y - \sin x e^{2y} + 2}{x^2 \cos y - 2\cos x e^{2y}}$