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To solve this problem, we need to do the following:

1. **Find the slope of the tangent line** to the curve \( y = \frac{2}{x} \) at the point \( \left( 6, \frac{1}{3} \right) \).
2. **Find the equation of the tangent line** in the form \( y = mx + b \).

### Step 1: Find the slope of the tangent line
The slope of the tangent line at a point on the curve is given by the derivative of the function at that point. The curve is:
\[
y = \frac{2}{x}
\]
To find the derivative \( \frac{dy}{dx} \), we use the power rule for differentiation:
\[
y = 2x^{-1} \quad \Rightarrow \quad \frac{dy}{dx} = -2x^{-2} = \frac{-2}{x^2}
\]
Now, substitute \( x = 6 \) into the derivative to find the slope at \( x = 6 \):
\[
\text{slope} = \frac{-2}{6^2} = \frac{-2}{36} = -\frac{1}{18}
\]
So, the slope of the tangent line is \( -\frac{1}{18} \).

### Step 2: Find the equation of the tangent line
We now know that the slope \( m = -\frac{1}{18} \). The equation of the tangent line can be written as:
\[
y = mx + b
\]
We need to find \( b \), the y-intercept. We can use the point \( \left( 6, \frac{1}{3} \right) \) which lies on the tangent line. Substituting the values \( x = 6 \), \( y = \frac{1}{3} \), and \( m = -\frac{1}{18} \) into the equation \( y = mx + b \):
\[
\frac{1}{3} = -\frac{1}{18}(6) + b
\]
Simplifying:
\[
\frac{1}{3} = -\frac{6}{18} + b
\]
\[
\frac{1}{3} = -\frac{1}{3} + b
\]
Now, solve for \( b \):
\[
b = \frac{1}{3} + \frac{1}{3} = \frac{2}{3}
\]

### Final Answers:
- The slope of the tangent line is \( -\frac{1}{18} \).
- The equation of the tangent line is \( y = -\frac{1}{18}x + \frac{2}{3} \).

Would you like a further breakdown or more details on any of the steps?

Here are 5 related questions to consider:
1. How do you find the derivative of a more complex function using the quotient rule?
2. How would the process change if the curve was a polynomial instead of a rational function?
3. Can the tangent line equation be rearranged in a different form (e.g., standard form)?
4. What does the slope of the tangent line represent in terms of the curve's behavior at the point?
5. How do you interpret the y-intercept of the tangent line in this context?

**Tip:** When finding the tangent line, always verify that the point you're using lies on the original curve to avoid errors.

The slope of the tangent line to the curve y = 2/x at the point (6, 1/3) is:

Learn how to find the slope and equation of the tangent line to the curve y = 2/x at the point (6, 1/3). This calculus problem covers derivatives and point-slope form, ideal for Grades 11-12.

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