To find the derivative of the function \( f(x) = 2x^2 - 3x + 1 \), we apply the rules of differentiation to each term separately.

1. The derivative of \( 2x^2 \) is calculated using the power rule \( \frac{d}{dx}(x^n) = nx^{n-1} \):
   \[
   \frac{d}{dx}(2x^2) = 2 \cdot 2x^{2-1} = 4x
   \]

2. The derivative of \( -3x \) is:
   \[
   \frac{d}{dx}(-3x) = -3
   \]

3. The derivative of a constant \( 1 \) is:
   \[
   \frac{d}{dx}(1) = 0
   \]

Combining these results, we get the derivative of the function \( f(x) \):
\[
f'(x) = 4x - 3
\]

Would you like more details or have any questions?

Here are some related questions to expand your understanding:

1. What is the second derivative of \( f(x) = 2x^2 - 3x + 1 \)?
2. How do you find the critical points of the function \( f(x) = 2x^2 - 3x + 1 \)?
3. What is the integral of \( f(x) = 2x^2 - 3x + 1 \)?
4. How do you determine the concavity of \( f(x) = 2x^2 - 3x + 1 \)?
5. What are the roots of the equation \( 2x^2 - 3x + 1 = 0 \)?
6. How do you find the vertex of the parabola given by \( f(x) = 2x^2 - 3x + 1 \)?
7. What is the slope of the tangent line to the curve \( f(x) = 2x^2 - 3x + 1 \) at \( x = 1 \)?
8. How does the graph of \( f(x) = 2x^2 - 3x + 1 \) look like?

**Tip:** When finding derivatives, always remember the basic differentiation rules such as the power rule, product rule, and chain rule.

Discover how to find the derivative of the function 2x^2 - 3x + 1 with a detailed step-by-step solution. This problem covers differentiation concepts, including the power rule and constant rule. Suitable for advanced high school students and beyond.

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