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We are given the following function to differentiate:

\[
y = \log_3\left(\left(\frac{3x^4}{5x^3 + 3}\right)^3\right)
\]

To find the derivative of this function, we can follow these steps:

### Step 1: Apply the change of base formula for logarithms
We can simplify the expression by changing the base of the logarithm from base 3 to base \( e \). Using the change of base formula:
\[
\log_b(x) = \frac{\ln(x)}{\ln(b)}
\]
we rewrite the given function as:
\[
y = \frac{\ln\left(\left(\frac{3x^4}{5x^3 + 3}\right)^3\right)}{\ln(3)}
\]

### Step 2: Simplify the logarithmic expression
Now, apply the logarithm power rule, \( \ln(a^b) = b \ln(a) \), to bring the exponent 3 in front:
\[
y = \frac{3 \ln\left(\frac{3x^4}{5x^3 + 3}\right)}{\ln(3)}
\]

### Step 3: Apply the logarithm quotient rule
Next, apply the logarithmic rule for division, \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \):
\[
y = \frac{3}{\ln(3)} \left[\ln(3x^4) - \ln(5x^3 + 3)\right]
\]

### Step 4: Differentiate each term
Now, differentiate the expression term by term. Let’s differentiate each logarithmic term using the chain rule.

- For \( \ln(3x^4) \):
  \[
  \frac{d}{dx} \ln(3x^4) = \frac{1}{3x^4} \cdot \frac{d}{dx}(3x^4) = \frac{1}{3x^4} \cdot 12x^3 = \frac{4}{x}
  \]

- For \( \ln(5x^3 + 3) \):
  \[
  \frac{d}{dx} \ln(5x^3 + 3) = \frac{1}{5x^3 + 3} \cdot \frac{d}{dx}(5x^3 + 3) = \frac{1}{5x^3 + 3} \cdot 15x^2 = \frac{15x^2}{5x^3 + 3}
  \]

### Step 5: Put everything together
The derivative of the entire function is:
\[
\frac{dy}{dx} = \frac{3}{\ln(3)} \left(\frac{4}{x} - \frac{15x^2}{5x^3 + 3}\right)
\]

This is the derivative of the given function.

Would you like further details or have any questions on these steps?

### Related Questions:
1. How do you apply the chain rule to logarithmic functions?
2. What is the change of base formula and how is it used in differentiation?
3. How can you differentiate a function involving a quotient of polynomials?
4. What is the logarithmic power rule and when should it be used?
5. How do you simplify complex logarithmic expressions before differentiating?

### Tip:
Always simplify the logarithmic expression as much as possible before differentiating, as this can make the derivative calculation easier.

Learn how to differentiate the logarithmic function y = log_3((3x^4 / (5x^3 + 3))^3) using logarithmic properties, the change of base formula, the chain rule, and the quotient rule. This solution walks through key steps such as simplifying the logarithmic expression and applying differentiation rules for complex functions.

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