To differentiate the implicit equation $xy^2 + 3y^3 = 5\cos(4x) + 2e^{2x} + \ln y$ with respect to $x$, we’ll apply implicit differentiation, treating $y$ as a function of $x$ (i.e., $y = y(x)$).

1. Differentiate each term on both sides with respect to $x$:

   $\frac{d}{dx} \left( xy^2 + 3y^3 \right) = \frac{d}{dx} \left( 5\cos(4x) + 2e^{2x} + \ln y \right)$

2. Differentiate each term on the left side:

   • For $xy^2$: Use the product rule: $\frac{d}{dx}(xy^2) = x \cdot \frac{d}{dx}(y^2) + y^2 \cdot \frac{d}{dx}(x) = x \cdot (2y \cdot y') + y^2 = 2xy y' + y^2$

   • For $3y^3$: $\frac{d}{dx}(3y^3) = 3 \cdot 3y^2 \cdot y' = 9y^2 y'$

   So, the derivative of the left side is: $2xy y' + y^2 + 9y^2 y'$

3. Differentiate each term on the right side:

   • For $5\cos(4x)$: $\frac{d}{dx}(5\cos(4x)) = 5 \cdot (-\sin(4x)) \cdot 4 = -20\sin(4x)$

   • For $2e^{2x}$: $\frac{d}{dx}(2e^{2x}) = 2 \cdot e^{2x} \cdot 2 = 4e^{2x}$

   • For $\ln y$: $\frac{d}{dx}(\ln y) = \frac{1}{y} \cdot y' = \frac{y'}{y}$

   Thus, the derivative of the right side is: $-20\sin(4x) + 4e^{2x} + \frac{y'}{y}$

4. Combine terms:

   We now have: $2xy y' + y^2 + 9y^2 y' = -20\sin(4x) + 4e^{2x} + \frac{y'}{y}$

5. Solve for $y'$:

   Collect all $y'$ terms on one side: $y' (2xy + 9y^2 - \frac{1}{y}) = -20\sin(4x) + 4e^{2x} - y^2$

   Finally, solving for $y'$: $y' = \frac{-20\sin(4x) + 4e^{2x} - y^2}{2xy + 9y^2 - \frac{1}{y}}$

The derivative $y'$ is: $y' = \frac{-20\sin(4x) + 4e^{2x} - y^2}{2xy + 9y^2 - \frac{1}{y}}$

Would you like further clarification on any part?

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Here are five related questions you might find helpful:

1. How would the differentiation change if $y$ were considered independent of $x$?
2. What are the product and chain rules for derivatives?
3. How do you differentiate trigonometric functions?
4. How do you solve implicit differentiation problems with natural logarithms?
5. What are the key steps for finding derivatives involving exponential functions?

Tip: Always apply the chain rule carefully when differentiating functions that include $y$ as a function of $x$.