We are given the function:

$f(x) = \frac{3x^3 + 8x^2}{x}$

Step 1: Simplify the function

We can simplify $f(x)$ by dividing each term in the numerator by $x$:

$f(x) = \frac{3x^3}{x} + \frac{8x^2}{x}$ $f(x) = 3x^2 + 8x$

Step 2: Differentiate the simplified function

Now, let's find the derivative of $f(x) = 3x^2 + 8x$ using the power rule:

• The derivative of $3x^2$ is $6x$ (since $\frac{d}{dx} [x^n] = n \cdot x^{n-1}$).
• The derivative of $8x$ is $8$.

Thus, the derivative $f'(x)$ is:

$f'(x) = 6x + 8$

Final Answer:

$f'(x) = 6x + 8$

Would you like more details or have any questions?

Here are 5 questions that build upon this:

1. How do you apply the quotient rule to more complex functions?
2. What is the general power rule for derivatives?
3. How do you differentiate functions involving both $x$ and constants?
4. Can the product rule be applied here instead of simplifying first?
5. What is the derivative of a function with higher-order polynomials?

Tip: Simplifying functions before differentiating can make the process much easier.