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We are tasked with differentiating the function \( f(x) = \frac{x^3 + 1}{4x^2 + 1} \) using the quotient rule.

### The Quotient Rule:
For two functions \( u(x) \) and \( v(x) \), where \( f(x) = \frac{u(x)}{v(x)} \), the quotient rule is given by:
\[
f'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2}
\]
In this case:
- \( u(x) = x^3 + 1 \)
- \( v(x) = 4x^2 + 1 \)

Now, we calculate the derivatives of \( u(x) \) and \( v(x) \):

1. \( u'(x) = \frac{d}{dx}(x^3 + 1) = 3x^2 \)
2. \( v'(x) = \frac{d}{dx}(4x^2 + 1) = 8x \)

### Apply the Quotient Rule:
\[
f'(x) = \frac{(4x^2 + 1)(3x^2) - (x^3 + 1)(8x)}{(4x^2 + 1)^2}
\]

Now, simplify the numerator:
\[
(4x^2 + 1)(3x^2) = 12x^4 + 3x^2
\]
\[
(x^3 + 1)(8x) = 8x^4 + 8x
\]
Thus, the numerator becomes:
\[
(12x^4 + 3x^2) - (8x^4 + 8x) = 4x^4 + 3x^2 - 8x
\]

So, the derivative is:
\[
f'(x) = \frac{4x^4 + 3x^2 - 8x}{(4x^2 + 1)^2}
\]

This is the simplified form of the derivative.

Would you like further clarification or details on any part of this solution?

### Related questions:
1. How does the quotient rule differ from the product rule?
2. Can we simplify the quotient rule further in specific cases?
3. What happens when \( v(x) = 1 \)? How does the quotient rule reduce?
4. How does the chain rule apply in more complex functions?
5. Can you differentiate a more complex rational function using the quotient rule?

**Tip**: Always simplify the terms inside the numerator before attempting to cancel terms when using the quotient rule!

Differentiate f(x) = (x^3 + 1) / (4x^2 + 1) using the quotient rule

Learn how to differentiate the function f(x) = (x^3 + 1) / (4x^2 + 1) using the quotient rule. This solution involves applying the quotient rule, simplifying the derivative of polynomials, and understanding core concepts of calculus. Suitable for high school and undergraduate calculus students.

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