To find the derivative of the function

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

we will use the quotient rule for differentiation. The quotient rule is given by:

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

where $u(x)$ is the numerator and $v(x)$ is the denominator of the fraction. So in our case:

• $u(x) = x^4 - x^2 + 1$
• $v(x) = x^2 + \sqrt{3}x + 1$

Now, we compute the derivatives of $u(x)$ and $v(x)$:

• $u'(x) = 4x^3 - 2x$
• $v'(x) = 2x + \sqrt{3}$

Using the quotient rule, the derivative is:

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

Next, I will simplify the expression.The derivative of the function

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

is:

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

This is the simplified form of the first derivative.

Would you like more details on the steps, or do you have any specific questions?

Here are five follow-up questions:

1. How can you apply the quotient rule to more complex functions?
2. What are the key differences between the quotient and product rules in differentiation?
3. How does simplifying derivatives help in practical applications?
4. Can this function be simplified further for specific values of $x$?
5. How would this derivative behave as $x \to \infty$?

Tip: Always verify the quotient rule by checking both the numerator and denominator derivatives separately before combining them.