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The derivative of the given function \( y = \log_3 \left( \frac{3x^4}{5x^3 + 3} \right)^3 \) with respect to \( x \) is:

\[
\frac{(15x^3 + 36) \cdot \log_3 \left( \frac{3x^4}{5x^3 + 3} \right)^2}{x \cdot (5x^3 + 3) \cdot (\log(3))^3}
\]

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Here are five related questions to deepen your understanding:

1. What is the derivative of a general logarithmic function \( \log_a(f(x)) \)?
2. How do you apply the chain rule when differentiating functions with logarithms?
3. What are some common properties of logarithms that simplify differentiation?
4. How can you use logarithmic differentiation for more complex products or quotients?
5. How does changing the base of a logarithm affect the derivative?

**Tip**: When working with logarithmic derivatives, converting to natural logarithms \( \ln(x) \) can sometimes simplify the process.

Find the derivative of y = log_3 [(3x^4)/(5x^3 + 3)]^3

Learn how to find the derivative of the logarithmic function y = log_3 [(3x^4)/(5x^3 + 3)]^3. This solution applies key concepts such as the chain rule, quotient rule, and logarithmic differentiation, suitable for advanced math students in Grades 11-12.

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