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We are given the function \( f(x) = \frac{1}{3 + x^2} \), and it is known that the slope of the graph of \( y = f(x) \) at \( x = 2 \) is \( -\frac{4}{49} \). We are tasked with finding the equation of the tangent line at this point.

### Step 1: Tangent Line Equation
The equation of the tangent line can be written in point-slope form:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope of the tangent line, and \( (x_1, y_1) \) is the point of tangency.

### Step 2: Find \( y_1 \) at \( x = 2 \)
We know that \( x_1 = 2 \). To find \( y_1 \), we substitute \( x = 2 \) into \( f(x) \):

\[
f(2) = \frac{1}{3 + 2^2} = \frac{1}{3 + 4} = \frac{1}{7}
\]

Thus, \( y_1 = \frac{1}{7} \).

### Step 3: Use Slope and Point
The slope \( m \) is given as \( -\frac{4}{49} \). Now we can substitute into the point-slope form:
\[
y - \frac{1}{7} = -\frac{4}{49}(x - 2)
\]

### Step 4: Simplify the Equation
Distribute the slope on the right-hand side:
\[
y - \frac{1}{7} = -\frac{4}{49}x + \frac{8}{49}
\]

Now, add \( \frac{1}{7} \) to both sides. Since \( \frac{1}{7} = \frac{7}{49} \), this gives:
\[
y = -\frac{4}{49}x + \frac{8}{49} + \frac{7}{49}
\]
\[
y = -\frac{4}{49}x + \frac{15}{49}
\]

### Final Answer:
The equation of the tangent line is:
\[
y = -\frac{4}{49}x + \frac{15}{49}
\]

Would you like further details on any step? Here are five related questions to deepen understanding:

1. How do we compute the derivative of \( f(x) = \frac{1}{3 + x^2} \) using the quotient rule?
2. How can we verify the given slope \( -\frac{4}{49} \) using the derivative?
3. How does the point-slope form of the equation of a line compare to the slope-intercept form?
4. What role does the derivative play in determining the tangent line to a function?
5. How would the tangent line change if the slope were positive at \( x = 2 \)?

**Tip:** When solving for the equation of a tangent line, always ensure the point and slope are calculated correctly before substituting into the point-slope form.

For f(x) = 1 / (3 + x^2), the slope of the graph of y = f(x) is known to be -4 / 49 at the point with x-coordinate 2. Find the equation of the tangent line at that point.

This problem explains how to find the equation of the tangent line to the curve f(x) = 1 / (3 + x^2) at x = 2, where the slope is given as -4 / 49. Learn about differentiation and applying the point-slope form to determine the equation of the tangent line.

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