1. **State the problem:** We need to find $\frac{dy}{dx}$ for the implicit equation $$xy^2 + \cos y = e^{3x} - 4 \tan y.$$\n\n2. **Recall the rules:** We will use implicit differentiation, differentiating both sides with respect to $x$. Remember that $y$ is a function of $x$, so when differentiating terms involving $y$, use the chain rule and multiply by $\frac{dy}{dx}$.\n\n3. **Differentiate each term:**\n- For $xy^2$, use the product rule: $$\frac{d}{dx}(xy^2) = x \frac{d}{dx}(y^2) + y^2 \frac{d}{dx}(x) = x(2y \frac{dy}{dx}) + y^2(1) = 2xy \frac{dy}{dx} + y^2.$$\n- For $\cos y$, $$\frac{d}{dx}(\cos y) = -\sin y \frac{dy}{dx}.$$\n- For $e^{3x}$, $$\frac{d}{dx}(e^{3x}) = 3e^{3x}.$$\n- For $-4 \tan y$, $$\frac{d}{dx}(-4 \tan y) = -4 \sec^2 y \frac{dy}{dx}.$$\n\n4. **Write the differentiated equation:**\n$$2xy \frac{dy}{dx} + y^2 - \sin y \frac{dy}{dx} = 3e^{3x} - 4 \sec^2 y \frac{dy}{dx}.$$\n\n5. **Group terms with $\frac{dy}{dx}$ on one side:**\n$$2xy \frac{dy}{dx} - \sin y \frac{dy}{dx} + 4 \sec^2 y \frac{dy}{dx} = 3e^{3x} - y^2.$$\n\n6. **Factor out $\frac{dy}{dx}$:**\n$$\left(2xy - \sin y + 4 \sec^2 y\right) \frac{dy}{dx} = 3e^{3x} - y^2.$$\n\n7. **Solve for $\frac{dy}{dx}$:**\n$$\frac{dy}{dx} = \frac{3e^{3x} - y^2}{2xy - \sin y + 4 \sec^2 y}.$$