If `y=ax^2 +b `, then `(dy)/(dx)` at `x=2` is equal to :

Options

A. `4a`
B. `3a`
C. `2a`
D. None of these

Answer

To find \(\frac{dy}{dx}\) at \(x=2\) for the function \(y = ax^2 + b\), we will follow these steps:

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

### Step 2: Apply the power rule
Using the power rule of differentiation, where \(\frac{d}{dx}(x^n) = nx^{n-1}\), we differentiate each term:
- The derivative of \(ax^2\) is \(2ax\) (since \(a\) is a constant).
- The derivative of \(b\) (a constant) is \(0\).

So, we have:
\[ \frac{dy}{dx} = 2ax + 0 = 2ax \]

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

### Conclusion
Thus, the value of \(\frac{dy}{dx}\) at \(x = 2\) is:
\[ \frac{dy}{dx} \bigg|_{x=2} = 4a \]