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To solve the problem, we need to find the derivative \( y' \) of the function given by:

\[
y = 3x^3 - 5x^2 + 8x - 1
\]

### Step 1: Differentiate each term of the function
The derivative of each term is calculated as follows:

1. For \( 3x^3 \), using the power rule \( \frac{d}{dx} [x^n] = nx^{n-1} \):
   \[
   \frac{d}{dx} [3x^3] = 3 \cdot 3x^{3-1} = 9x^2
   \]

2. For \( -5x^2 \), applying the same rule:
   \[
   \frac{d}{dx} [-5x^2] = -5 \cdot 2x^{2-1} = -10x
   \]

3. For \( 8x \), the derivative is simply:
   \[
   \frac{d}{dx} [8x] = 8
   \]

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

### Step 2: Combine the derivatives
Putting it all together:

\[
y' = 9x^2 - 10x + 8
\]

### Conclusion
The correct answer from the given options is:

\[
\boxed{9x^2 - 10x + 8}
\]

Would you like to go over any details or have any questions?

### Related Questions:

1. What is the derivative of \( y = 5x^4 - 3x^2 + 2x - 7 \)?
2. How do you find the second derivative of \( y = 4x^3 - 6x^2 + 2x - 5 \)?
3. What is the integral of the function \( y = 3x^2 - 4x + 7 \)?
4. How do you apply the power rule for differentiation?
5. What is the derivative of \( y = 7x^5 - x^3 + 4x^2 - 8x + 1 \)?
6. How does one differentiate a product of functions?
7. What is the role of constants in differentiation?
8. How do you find the derivative of a polynomial function?

**Tip:** When differentiating a polynomial, remember that the derivative of a constant term is always zero.

Learn how to find the derivative of the polynomial function 3x^3 - 5x^2 + 8x - 1 step-by-step. This problem covers differentiation of polynomials using the power rule. Suitable for advanced high school students and college freshmen.

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