To differentiate the function $f(x) = \frac{x^3 + 1}{4x^2 + 1}$ using the quotient rule, we'll follow these steps:

The quotient rule states:

$\frac{d}{dx}\left( \frac{u(x)}{v(x)} \right) = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{v(x)^2}$

Where:

• $u(x) = x^3 + 1$
• $v(x) = 4x^2 + 1$

Step 1: Differentiate $u(x)$ and $v(x)$

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

Step 2: Apply the quotient rule

Using the formula for the quotient rule:

$f'(x) = \frac{(4x^2 + 1) \cdot (3x^2) - (x^3 + 1) \cdot (8x)}{(4x^2 + 1)^2}$

Step 3: Simplify the numerator

Expand the terms in the numerator:

$\text{Numerator} = (4x^2 + 1)(3x^2) - (x^3 + 1)(8x)$ $= (12x^4 + 3x^2) - (8x^4 + 8x)$ $= 12x^4 + 3x^2 - 8x^4 - 8x$ $= (12x^4 - 8x^4) + 3x^2 - 8x$ $= 4x^4 + 3x^2 - 8x$

Step 4: Final expression

Now substitute the simplified numerator back into the quotient rule formula:

$f'(x) = \frac{4x^4 + 3x^2 - 8x}{(4x^2 + 1)^2}$

This is the derivative of $f(x)$ using the quotient rule.

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you apply the product rule in differentiation?
2. Can you explain when the chain rule is used in differentiation?
3. What is the difference between the quotient rule and the product rule?
4. How do you handle higher-order derivatives for quotient functions?
5. Can the function $f(x)$ be simplified before applying the quotient rule?

Tip: Always simplify the function as much as possible before applying differentiation rules to reduce calculation errors.