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Let's solve problem number 10 step by step.

You are asked to find the derivative of the function:

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

### Step 1: Simplify the Function
We can first simplify the function by dividing each term in the numerator by \(x\):

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

This simplifies to:

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

### Step 2: Rewrite the Function in Terms of Powers of \(x\)
The term \(\frac{2}{x}\) can be rewritten as \(2x^{-1}\), so the function becomes:

\[
f(x) = x^3 - 5 + 2x^{-1}
\]

### Step 3: Differentiate the Function
Now, we can take the derivative of each term using basic rules of differentiation:

- The derivative of \(x^3\) is \(3x^2\)
- The derivative of \(-5\) is \(0\)
- The derivative of \(2x^{-1}\) is \(-2x^{-2}\)

Thus, the derivative of \(f(x)\) is:

\[
f'(x) = 3x^2 - 2x^{-2}
\]

### Step 4: Final Answer
The derivative of the function is:

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

Would you like more details on any part of the solution?

### Follow-up questions:
1. How would you apply the power rule in differentiation to more complex functions?
2. What is the general rule for differentiating a function with negative exponents?
3. Can you explain why the constant term \(-5\) becomes 0 after differentiation?
4. How would you check if the derivative you've calculated is correct using an alternative method?
5. How can you interpret the derivative \(f'(x)\) in terms of the slope of the original function?

#### Tip:
When differentiating functions involving fractions, try to rewrite all terms as powers of \(x\) to make the process more straightforward.

Find the derivative of f(x) = (x^4 - 5x + 2) / x.

Learn how to find the derivative of the function f(x) = (x^4 - 5x + 2)/x using the power rule and simplifying rational functions. This problem focuses on key calculus concepts such as the power rule and differentiation of functions with negative exponents, suitable for students in Grades 11-12.

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