We are tasked with finding the equation of the tangent line to the function $f(x) = \frac{7}{6 + x^2}$ at $x = 6$.

Steps to Solve:

1. Find the value of $f(6)$: This gives us the point on the curve where the tangent line touches.

   $f(6) = \frac{7}{6 + 6^2} = \frac{7}{6 + 36} = \frac{7}{42} = \frac{1}{6}$

2. Find $f'(x)$, the derivative of $f(x)$: We'll use the quotient rule to differentiate. The quotient rule states:

   $f'(x) = \frac{d}{dx} \left( \frac{7}{6 + x^2} \right) = \frac{(6 + x^2) \cdot 0 - 7 \cdot 2x}{(6 + x^2)^2} = \frac{-14x}{(6 + x^2)^2}$

3. Evaluate $f'(6)$, the slope of the tangent line at $x = 6$:

   $f'(6) = \frac{-14(6)}{(6 + 6^2)^2} = \frac{-84}{(6 + 36)^2} = \frac{-84}{42^2} = \frac{-84}{1764} = -\frac{7}{147} = -\frac{1}{21}$

4. Equation of the tangent line: The equation of the tangent line is given by the point-slope form:

   $y - f(6) = f'(6)(x - 6)$

   Substituting $f(6) = \frac{1}{6}$ and $f'(6) = -\frac{1}{21}$:

   $y - \frac{1}{6} = -\frac{1}{21}(x - 6)$

   Simplifying the equation:

   $y - \frac{1}{6} = -\frac{1}{21}x + \frac{6}{21} = -\frac{1}{21}x + \frac{2}{7}$

   Finally, add $\frac{1}{6}$ to both sides to get the equation of the tangent line:

   $y = -\frac{1}{21}x + \frac{2}{7} + \frac{1}{6}$

   To combine the constants $\frac{2}{7}$ and $\frac{1}{6}$, find a common denominator:

   $\frac{2}{7} + \frac{1}{6} = \frac{12}{42} + \frac{7}{42} = \frac{19}{42}$

   Thus, the equation of the tangent line is:

   $y = -\frac{1}{21}x + \frac{19}{42}$

Would you like further details or have any questions about this process? Here are 5 related questions you might explore:

1. How is the quotient rule applied in general when differentiating functions?
2. What are the alternative methods to find the slope of a tangent line?
3. Can we always use the point-slope form to find the equation of a tangent line?
4. How do different types of functions (e.g., polynomials or trigonometric functions) affect the shape of their tangent lines?
5. How does the concavity of a function impact the behavior of its tangent line?

Tip: When combining fractions, always check if they can be simplified before adding, which makes calculations more manageable.