Find the derivative of the function.
f(x) = 
3x3 + 8x2
x


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.

Find the derivative of the function f(x) = (3x^3 + 8x^2) / x

Find the derivative of the rational function f(x) = (3x^3 + 8x^2) / x using simplification and the power rule. This step-by-step solution is ideal for students learning calculus concepts, including derivatives and polynomial functions. Suitable for Grades 11-12 or Calculus I.

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