If `y=2x^(3)+3x^(2)+6x+1`, then `(dy)/(dx)` will be `-`

Options

A. `6(x^(2)+x+1)`
B. `6(x^(2)+x_2)`
C. `6x^(2)+3x`
D. `(x^(2)+6x+1)`

Explanation

To find the derivative \( \frac{dy}{dx} \) of the function \( y = 2x^3 + 3x^2 + 6x + 1 \), we will apply the rules of differentiation step by step.

### Step 1: Identify the function
We have the function:
\[ y = 2x^3 + 3x^2 + 6x + 1 \]

### Step 2: Differentiate each term
We will differentiate each term of the function separately using the power rule of differentiation, which states that \( \frac{d}{dx}(x^n) = n \cdot x^{n-1} \).

1. Differentiate \( 2x^3 \):
\[ \frac{d}{dx}(2x^3) = 2 \cdot 3x^{3-1} = 6x^2 \]

2. Differentiate \( 3x^2 \):
\[ \frac{d}{dx}(3x^2) = 3 \cdot 2x^{2-1} = 6x \]

3. Differentiate \( 6x \):
\[ \frac{d}{dx}(6x) = 6 \cdot 1 = 6 \]

4. Differentiate the constant \( 1 \):
\[ \frac{d}{dx}(1) = 0 \]

### Step 3: Combine the derivatives
Now, we will combine the results from the differentiation of each term:
\[ \frac{dy}{dx} = 6x^2 + 6x + 6 \]

### Step 4: Factor the expression (optional)
We can factor out the common factor of \( 6 \):
\[ \frac{dy}{dx} = 6(x^2 + x + 1) \]

### Final Result
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = 6(x^2 + x + 1) \]