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To find the derivative of the given function, we can use the Quotient Rule. The function provided is:

\[
f(x) = \frac{5x^3 + 3}{x^4 - 2}
\]

The Quotient Rule states that for a function \(\frac{g(x)}{h(x)}\), the derivative \(f'(x)\) is:

\[
f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}
\]

Here, \(g(x) = 5x^3 + 3\) and \(h(x) = x^4 - 2\). Let's calculate the derivatives of \(g(x)\) and \(h(x)\):

\[
g'(x) = \frac{d}{dx}(5x^3 + 3) = 15x^2
\]
\[
h'(x) = \frac{d}{dx}(x^4 - 2) = 4x^3
\]

Now, apply the Quotient Rule:

\[
f'(x) = \frac{(15x^2)(x^4 - 2) - (5x^3 + 3)(4x^3)}{(x^4 - 2)^2}
\]

Let's simplify this expression:

\[
f'(x) = \frac{15x^2(x^4 - 2) - 4x^3(5x^3 + 3)}{(x^4 - 2)^2}
\]

Expanding both products:

\[
f'(x) = \frac{15x^6 - 30x^2 - (20x^6 + 12x^3)}{(x^4 - 2)^2}
\]

Combining like terms:

\[
f'(x) = \frac{15x^6 - 20x^6 - 12x^3 - 30x^2}{(x^4 - 2)^2}
\]
\[
f'(x) = \frac{-5x^6 - 12x^3 - 30x^2}{(x^4 - 2)^2}
\]

So, the derivative of the function is:

\[
f'(x) = \frac{-5x^6 - 12x^3 - 30x^2}{(x^4 - 2)^2}
\]

Would you like further details on any of the steps?

### Related Questions:
1. How does the Quotient Rule differ from the Product Rule in calculus?
2. What is the significance of each term in the Quotient Rule?
3. How would the derivative change if the denominator was \(x^4 + 2\) instead of \(x^4 - 2\)?
4. How do the signs of the terms in the numerator affect the overall derivative?
5. Can you apply the Chain Rule to this function? If so, how?

### Tip:
When applying the Quotient Rule, it's helpful to keep the numerator and denominator derivatives separate until the final step to avoid mistakes during expansion and simplification.

Learn how to find the derivative of the function f(x) = (5x^3 + 3) / (x^4 - 2) using the Quotient Rule. This problem covers calculus concepts, including derivatives and the application of the Quotient Rule formula. Suitable for college-level students.

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