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The problem asks to find \( \frac{dy}{dx} \) for the given function:

\[
y = \ln\left( \left( \frac{2x^3}{3x^2 + 5} \right)^4 \right)
\]

### Steps to find \( \frac{dy}{dx} \):

1. **Simplify the expression:**
   Use the logarithmic property \( \ln(a^b) = b \ln(a) \):
   \[
   y = 4 \ln \left( \frac{2x^3}{3x^2 + 5} \right)
   \]

2. **Differentiate using the chain rule and quotient rule:**
   The derivative of \( \ln(u) \) is \( \frac{1}{u} \cdot \frac{du}{dx} \), where \( u = \frac{2x^3}{3x^2 + 5} \).

   Let:
   \[
   u = \frac{2x^3}{3x^2 + 5}
   \]
   Now, use the quotient rule to differentiate \( u \):
   \[
   \frac{du}{dx} = \frac{(3x^2 + 5)(6x^2) - (2x^3)(6x)}{(3x^2 + 5)^2}
   \]
   Simplify the numerator:
   \[
   = \frac{(3x^2 + 5)(6x^2) - (12x^4)}{(3x^2 + 5)^2}
   \]
   Expand the terms:
   \[
   = \frac{18x^4 + 30x^2 - 12x^4}{(3x^2 + 5)^2}
   \]
   \[
   = \frac{6x^4 + 30x^2}{(3x^2 + 5)^2}
   \]

3. **Final expression for \( \frac{dy}{dx} \):**
   Now, applying the chain rule for \( y = 4 \ln(u) \):
   \[
   \frac{dy}{dx} = 4 \cdot \frac{1}{u} \cdot \frac{du}{dx}
   \]
   Substitute \( u = \frac{2x^3}{3x^2 + 5} \) and the derivative \( \frac{du}{dx} \):
   \[
   \frac{dy}{dx} = 4 \cdot \frac{1}{\frac{2x^3}{3x^2 + 5}} \cdot \frac{6x^4 + 30x^2}{(3x^2 + 5)^2}
   \]
   Simplify:
   \[
   \frac{dy}{dx} = 4 \cdot \frac{(3x^2 + 5)}{2x^3} \cdot \frac{6x^4 + 30x^2}{(3x^2 + 5)^2}
   \]
   Cancel terms:
   \[
   \frac{dy}{dx} = \frac{2(6x^4 + 30x^2)}{x^3(3x^2 + 5)}
   \]

   This is the derivative of the function.

Would you like further clarification or a detailed breakdown of any step?

Here are 5 related questions:
1. How do logarithmic properties simplify differentiation problems?
2. What is the general formula for the quotient rule in differentiation?
3. How can the chain rule be applied to nested functions like in this problem?
4. How would the result change if the exponent were different?
5. Can you simplify the derivative further in terms of common factors?

**Tip:** Always check if expressions can be simplified before differentiating—it often reduces the complexity of the derivative!

Find dy/dx if y = ln((2x^3 / (3x^2 + 5))^4)

Learn how to differentiate the logarithmic function y = ln((2x^3 / (3x^2 + 5))^4) using the chain rule and quotient rule. This problem covers advanced calculus concepts and provides a detailed step-by-step solution.

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