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The problem asks to find the equation of the tangent line to the graph of the function \( y = \frac{2x - 6}{2x + 6} \) at the point \( \left( 6, \frac{1}{3} \right) \).

### Steps to solve:

1. **Find the derivative**: We need the derivative \( \frac{dy}{dx} \), which represents the slope of the tangent line at any point \( x \). This can be found using the quotient rule:
   \[
   \frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2}
   \]
   where \( u(x) = 2x - 6 \) and \( v(x) = 2x + 6 \).

2. **Apply the quotient rule**:
   - \( u(x) = 2x - 6 \), so \( u'(x) = 2 \)
   - \( v(x) = 2x + 6 \), so \( v'(x) = 2 \)

   Using the quotient rule:
   \[
   \frac{dy}{dx} = \frac{(2x + 6)(2) - (2x - 6)(2)}{(2x + 6)^2}
   \]

3. **Simplify**:
   \[
   \frac{dy}{dx} = \frac{2(2x + 6) - 2(2x - 6)}{(2x + 6)^2}
   = \frac{4x + 12 - (4x - 12)}{(2x + 6)^2}
   = \frac{4x + 12 - 4x + 12}{(2x + 6)^2}
   = \frac{24}{(2x + 6)^2}
   \]

   So, \( \frac{dy}{dx} = \frac{24}{(2x + 6)^2} \).

4. **Evaluate the derivative at \( x = 6 \)**:
   \[
   \frac{dy}{dx} \bigg|_{x=6} = \frac{24}{(2(6) + 6)^2} = \frac{24}{(12 + 6)^2} = \frac{24}{18^2} = \frac{24}{324} = \frac{2}{27}
   \]

   Therefore, the slope of the tangent line at \( x = 6 \) is \( m = \frac{2}{27} \).

5. **Find the equation of the tangent line**:
   The equation of a line is given by \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is the point of tangency and \( m \) is the slope.

   Here, \( (x_1, y_1) = \left( 6, \frac{1}{3} \right) \) and \( m = \frac{2}{27} \), so the equation becomes:
   \[
   y - \frac{1}{3} = \frac{2}{27}(x - 6)
   \]

6. **Simplify the equation**:
   \[
   y = \frac{2}{27}(x - 6) + \frac{1}{3}
   \]
   Multiply both sides to eliminate fractions if desired.

This is the equation of the tangent line at \( \left( 6, \frac{1}{3} \right) \).

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Would you like further details or have any questions?

Here are 5 related questions:
1. How is the quotient rule derived?
2. What happens if the denominator in a rational function has higher complexity?
3. How do we interpret the derivative geometrically?
4. Why do we need to find the derivative to determine the tangent line?
5. How does a derivative behave when the function has discontinuities?

**Tip**: Always simplify the function as much as possible before applying differentiation rules; it reduces errors during calculation.

Find an equation of the tangent line to the graph of the function y = (2x - 6) / (2x + 6) at the point (6, 1/3).

Learn how to find the equation of the tangent line to the function y = (2x - 6) / (2x + 6) at the point (6, 1/3) using calculus. This problem involves the quotient rule for differentiation and calculating the slope of the tangent line.

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