We are tasked with differentiating the function:

\[
f(x) = (4x^2 - 6x + 2) \cdot x^{-\frac{1}{2}}
\]

### Step-by-Step Differentiation:

1. **Distribute** the \(x^{-\frac{1}{2}}\) across the terms inside the parentheses:

\[
f(x) = 4x^2 \cdot x^{-\frac{1}{2}} - 6x \cdot x^{-\frac{1}{2}} + 2 \cdot x^{-\frac{1}{2}}
\]

This gives us:

\[
f(x) = 4x^{2 - \frac{1}{2}} - 6x^{1 - \frac{1}{2}} + 2x^{-\frac{1}{2}}
\]

Which simplifies to:

\[
f(x) = 4x^{\frac{3}{2}} - 6x^{\frac{1}{2}} + 2x^{-\frac{1}{2}}
\]

2. **Differentiate term by term** using the power rule: \(\frac{d}{dx} [x^n] = n x^{n-1}\).

- For the first term: \( 4x^{\frac{3}{2}} \), the derivative is:
  \[
  \frac{d}{dx} \left( 4x^{\frac{3}{2}} \right) = 4 \cdot \frac{3}{2} x^{\frac{3}{2} - 1} = 6x^{\frac{1}{2}}
  \]

- For the second term: \( -6x^{\frac{1}{2}} \), the derivative is:
  \[
  \frac{d}{dx} \left( -6x^{\frac{1}{2}} \right) = -6 \cdot \frac{1}{2} x^{\frac{1}{2} - 1} = -3x^{-\frac{1}{2}}
  \]

- For the third term: \( 2x^{-\frac{1}{2}} \), the derivative is:
  \[
  \frac{d}{dx} \left( 2x^{-\frac{1}{2}} \right) = 2 \cdot -\frac{1}{2} x^{-\frac{1}{2} - 1} = -x^{-\frac{3}{2}}
  \]

3. **Combine the results**:

\[
f'(x) = 6x^{\frac{1}{2}} - 3x^{-\frac{1}{2}} - x^{-\frac{3}{2}}
\]

This is the derivative of the given function.

Would you like further explanation or examples? 

Here are 5 related questions you might find helpful:
1. How do you apply the product rule when differentiating?
2. What is the general power rule for differentiation?
3. How can you simplify expressions involving fractional exponents?
4. How do you integrate functions with fractional exponents?
5. How does the chain rule work in combination with the power rule?

**Tip**: When working with fractional exponents, remember that \( x^{1/n} \) represents the \(n\)-th root of \(x\).

This detailed solution walks through the differentiation of the function (4x^2 - 6x + 2) * x^(-1/2). Learn how to apply the power rule and manage fractional exponents in this step-by-step calculus tutorial, suitable for Grades 11-12 and early college students.

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