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To find the slope of the tangent line to the graph of \( f(x) = \sqrt{x + 6} \) at \( x = 3 \), we need to calculate the derivative of the function \( f(x) \), which gives the slope of the tangent line at any point \( x \).

### Step 1: Differentiate the function
Given \( f(x) = \sqrt{x + 6} \), we can rewrite this as \( f(x) = (x + 6)^{1/2} \).

The derivative of \( f(x) = (x + 6)^{1/2} \) is found using the chain rule:
\[
f'(x) = \frac{1}{2}(x + 6)^{-1/2} \cdot 1
\]
\[
f'(x) = \frac{1}{2\sqrt{x + 6}}
\]

### Step 2: Evaluate the derivative at \( x = 3 \)
Now, substitute \( x = 3 \) into the derivative:
\[
f'(3) = \frac{1}{2\sqrt{3 + 6}} = \frac{1}{2\sqrt{9}} = \frac{1}{2 \times 3} = \frac{1}{6}
\]

Thus, the slope of the tangent line at \( x = 3 \) is \( \frac{1}{6} \).

### Final Answer:
\[
m_{\text{tan}} = \frac{1}{6}
\]

Would you like any more details on this process? Here are five related questions you might find useful:

1. How is the chain rule applied to find the derivative of a composite function?
2. What is the geometric interpretation of the derivative at a point?
3. How does the slope of a tangent line relate to the rate of change of a function?
4. How can you find the equation of the tangent line once you know the slope?
5. What are the steps to find a derivative using the power rule?

**Tip**: The chain rule is crucial for differentiating composite functions like \( \sqrt{x + 6} \). Always break down the function into its outer and inner components when applying it.

Find the slope of the tangent line to the graph of f(x) = sqrt(x + 6) at x = 3.

Learn how to find the slope of the tangent line to the graph of f(x) = sqrt(x + 6) at x = 3. This calculus problem covers derivatives and the application of the chain rule, suitable for high school students studying calculus.

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