`y=x^(3)+2x^(2)+7x+8` then `(dy)/(dx)` will be-

Options

A. `3x^(2)+2x+15`
B. `3x^(2)+4x+7`
C. `x^(3)+2x^(2)+15`
D. `x^(3)+4x+7`

Explanation

To find the derivative of the function \( y = x^3 + 2x^2 + 7x + 8 \) with respect to \( x \), we will apply the rules of differentiation step by step.

### Step 1: Write down the function
We start with the function:
\[ y = x^3 + 2x^2 + 7x + 8 \]

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

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

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

3. Differentiate \( 7x \):
\[ \frac{d}{dx}(7x) = 7 \cdot \frac{d}{dx}(x) = 7 \cdot 1 = 7 \]

4. Differentiate the constant \( 8 \):
\[ \frac{d}{dx}(8) = 0 \]

### Step 3: Combine the derivatives
Now, we combine all the derivatives we calculated:
\[ \frac{dy}{dx} = 3x^2 + 4x + 7 + 0 \]

### Step 4: Simplify the expression
Thus, the final expression for the derivative is:
\[ \frac{dy}{dx} = 3x^2 + 4x + 7 \]

### Final Answer
The derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = 3x^2 + 4x + 7 \]

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