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The derivative of the function \( y = \frac{x^3}{(x^3 + 1)^5} \) with respect to \( x \) is:

\[
\frac{dy}{dx} = \frac{x^2 \cdot (3 - 12x^3)}{(x^3 + 1)^6}
\]

Would you like further clarification on any steps, or do you have additional questions? 

Here are five related questions:
1. How do you find the derivative of a quotient?
2. What is the chain rule, and how is it applied here?
3. What happens if \( x = 0 \) in this derivative?
4. How can you simplify complex fractions involving polynomials?
5. How would you apply this derivative in a real-world scenario?

**Tip:** Always remember to simplify derivatives where possible, as it can make further analysis much easier.

Derive the function: y = x^3 / (x^3 + 1)^5

Learn how to derive the function y = x^3 / (x^3 + 1)^5 using the quotient rule and chain rule in calculus. This detailed solution covers key differentiation techniques, including the application of the quotient rule and chain rule. Suitable for undergraduate calculus students.

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