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Let's solve the problem step by step.

### Given Function:
\[ f(x) = \sqrt{7x + 1} \]
You are asked to:
1. Find \( f'(a) \) when \( a = 5 \).

### Step 1: Find the derivative of \( f(x) \).

We will use the **chain rule** to differentiate \( f(x) = \sqrt{7x + 1} = (7x + 1)^{1/2} \).

Using the chain rule:
\[
f'(x) = \frac{1}{2} (7x + 1)^{-1/2} \cdot 7
\]
Simplifying:
\[
f'(x) = \frac{7}{2\sqrt{7x + 1}}
\]

### Step 2: Evaluate \( f'(a) \) when \( a = 5 \).

Substitute \( x = 5 \) into the derivative:
\[
f'(5) = \frac{7}{2\sqrt{7(5) + 1}} = \frac{7}{2\sqrt{36}} = \frac{7}{2 \times 6} = \frac{7}{12}
\]

So, \( f'(5) = \frac{7}{12} \).

Would you like more details or help with part b of the problem? Here are some follow-up questions for deeper understanding:

1. Can you describe how the chain rule was applied in this problem?
2. Why is it important to simplify the expression for \( f'(x) \) before plugging in values?
3. What is the geometrical interpretation of \( f'(a) \)?
4. How does the derivative relate to the slope of the tangent line at a point on a graph?
5. Could you explain how \( f'(a) \) helps find the equation of the tangent line?

**Tip:** Always check if functions like square roots can be simplified before applying differentiation rules to reduce complexity.

For the following function, find f'(a). Determine an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a. Function f(x) = sqrt(7x + 1), a = 5

Explore a step-by-step guide on calculating derivatives using the chain rule and determining the equation of the tangent line for the function f(x) = sqrt(7x + 1) at x = 5. This example delves into core calculus concepts, suitable for college students.

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