`x^(3) -2x^(2) +x+3`

Answer

To differentiate the function \( y = x^3 - 2x^2 + x + 3 \), we will apply the rules of differentiation step by step.

### Step 1: Write down the function
We start with the function:
\[ y = x^3 - 2x^2 + x + 3 \]

### Step 2: Differentiate each term
We will differentiate each term of the function separately.

1. **Differentiate \( x^3 \)**:
\[ \frac{d}{dx}(x^3) = 3x^{3-1} = 3x^2 \]

2. **Differentiate \( -2x^2 \)**:
\[ \frac{d}{dx}(-2x^2) = -2 \cdot 2x^{2-1} = -4x \]

3. **Differentiate \( x \)**:
\[ \frac{d}{dx}(x) = 1 \]

4. **Differentiate the constant \( 3 \)**:
\[ \frac{d}{dx}(3) = 0 \]

### Step 3: Combine the derivatives
Now, we combine all the derivatives we calculated:
\[ \frac{dy}{dx} = 3x^2 - 4x + 1 + 0 \]
Thus, we simplify it to:
\[ \frac{dy}{dx} = 3x^2 - 4x + 1 \]

### Final Answer
The derivative of the function \( y = x^3 - 2x^2 + x + 3 \) is:
\[ \frac{dy}{dx} = 3x^2 - 4x + 1 \]

---