The derivative of the function $f(x) = \sqrt{7x + 4}$ is:

$f'(x) = \frac{7}{2\sqrt{7x + 4}}$

Let me know if you have any questions, or if you'd like me to proceed with part (b) of the problem!

Here are some related questions for further exploration:

1. What is the meaning of the derivative in terms of the slope of the tangent line?
2. How do you find the equation of a tangent line at a given point?
3. What happens to the derivative as $x$ increases?
4. How would you handle finding the derivative of a more complex square root function?
5. Can you interpret the behavior of the graph of $f(x)$ from the derivative?

Tip: Always simplify your derivative as much as possible to make calculations easier!