The problem asks to differentiate the function $y = \sqrt{4x + 7}$ with respect to $x$.

1. Problem Description

The problem asks to differentiate the function $y = \sqrt{4x + 7}$ with respect to $x$.

2. Solution Steps

We can rewrite the function as $y = (4x + 7)^{1/2}$.

Using the chain rule, we have

$\frac{dy}{dx} = \frac{d}{dx} (4x + 7)^{1/2}$

$\frac{dy}{dx} = \frac{1}{2} (4x + 7)^{(1/2) - 1} \cdot \frac{d}{dx} (4x + 7)$

$\frac{dy}{dx} = \frac{1}{2} (4x + 7)^{-1/2} \cdot (4)$

$\frac{dy}{dx} = 2 (4x + 7)^{-1/2}$

$\frac{dy}{dx} = \frac{2}{\sqrt{4x + 7}}$

3. Final Answer

The derivative of $y = \sqrt{4x + 7}$ is $\frac{dy}{dx} = \frac{2}{\sqrt{4x + 7}}$.