If y = ax² + b, then dy/dx at x = 2 is equal to ax

The correct option is 4a.

To find the derivative of the function, we begin with the given equation:
$$y=a{x}^2+b$$

We differentiate both sides with respect to x. Using the power rule of differentiation, which states that the derivative of $x^n$ with respect to x is $n{x}^{n-1}$, and knowing that the derivative of a constant (like b) is zero, we get:
$$\frac{dy}{dx}=\frac{d}{dx}(a{x}^2+b)$$
$$\frac{dy}{dx}=a\cdot \left(2x\right)+0$$
$$\frac{dy}{dx}=2ax$$

Now, we need to evaluate this derivative at the specific value $x=2$. By substituting $x=2$ into our derivative expression, we obtain:
$${[\frac{dy}{dx}}_{x=2}=2a\left(2\right)$$
$${[\frac{dy}{dx}}_{x=2}=4a$$