If `y=2x^2+12x+35` , then `dy/dx` at x=3

Options

A. 12
B. 28
C. 32
D. 24

Answer

To find the derivative \( \frac{dy}{dx} \) of the function \( y = 2x^2 + 12x + 35 \) at \( x = 3 \), we will follow these steps:

### Step 1: Differentiate the function
We start with the function:
\[ y = 2x^2 + 12x + 35 \]
To find \( \frac{dy}{dx} \), we differentiate each term with respect to \( x \).

- The derivative of \( 2x^2 \) is \( 4x \) (using the power rule).
- The derivative of \( 12x \) is \( 12 \).
- The derivative of the constant \( 35 \) is \( 0 \).

Putting it all together, we have:
\[ \frac{dy}{dx} = 4x + 12 \]

### Step 2: Substitute \( x = 3 \)
Now, we need to evaluate \( \frac{dy}{dx} \) at \( x = 3 \):
\[ \frac{dy}{dx} \bigg|_{x=3} = 4(3) + 12 \]

### Step 3: Calculate the value
Calculating this gives:
\[ \frac{dy}{dx} \bigg|_{x=3} = 12 + 12 = 24 \]

### Final Result
Thus, the value of \( \frac{dy}{dx} \) at \( x = 3 \) is:
\[ \frac{dy}{dx} = 24 \]