Here’s a question that involves derivatives using all four rules you mentioned: the product rule, power rule, quotient rule, and chain rule.

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Problem:

Find the derivative of the function:

$f(x) = \frac{(3x^2 + 2)(x^3 - 4)}{(x^4 + 1)^2}$

Solution involves:

1. Product Rule: Applied to $(3x^2 + 2)(x^3 - 4)$.
2. Power Rule: Used for individual powers of $x$ in the product.
3. Quotient Rule: Used for the division by $(x^4 + 1)^2$.
4. Chain Rule: Applied for the squared term $(x^4 + 1)^2$ in the denominator.

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Would you like me to break down the solution step by step, or do you want to try solving it first? Feel free to ask if you need more clarification or if you have another question!

Here are some follow-up questions for you:

1. How does the product rule differ from the quotient rule in terms of derivative calculations?
2. Can you identify when to apply the chain rule within composite functions?
3. What is the result of applying the power rule to a function like $x^n$, where $n$ is a constant?
4. How would you apply the quotient rule if both the numerator and denominator involved polynomial expressions?
5. What happens if you forget to apply the chain rule correctly in a derivative problem?

Tip: Remember that the chain rule is crucial for nested functions, so always check if one function is inside another before differentiating!