find the equation of the tangent line $$ \frac{8}{3x^2} $$ at x=-4

We are tasked with finding the equation of the tangent line to the function $ f(x) = \frac{8}{3x^2} $ at $ x = -4 $.

To find the equation of the tangent line, we need two key things:

1. **The slope of the tangent line**, which is given by the derivative of the function $ f'(x) $ at $ x = -4 $.
2. **The point of tangency**, which includes the coordinates $ (x_0, y_0) $ where $ x_0 = -4 $ and $ y_0 = f(-4) $.

### Step 1: Find the derivative $ f'(x) $
Given $ f(x) = \frac{8}{3x^2} $, we can rewrite this as:
\[
f(x) = \frac{8}{3} x^{-2}
\]
Now, differentiate using the power rule:
\[
f'(x) = \frac{8}{3} \cdot (-2) x^{-3} = -\frac{16}{3} x^{-3} = -\frac{16}{3x^3}
\]

### Step 2: Calculate $ f'(-4) $
Now substitute $ x = -4 $ into the derivative:
\[
f'(-4) = -\frac{16}{3(-4)^3} = -\frac{16}{3(-64)} = \frac{16}{192} = \frac{1}{12}
\]
Thus, the slope of the tangent line at $ x = -4 $ is $ \frac{1}{12} $.

### Step 3: Find the point $ (x_0, y_0) $
Next, find $ f(-4) $:
\[
f(-4) = \frac{8}{3(-4)^2} = \frac{8}{3 \times 16} = \frac{8}{48} = \frac{1}{6}
\]
So, the point of tangency is $ (-4, \frac{1}{6}) $.

### Step 4: Write the equation of the tangent line
The equation of the tangent line in point-slope form is:
\[
y - y_0 = m(x - x_0)
\]
Substitute $ m = \frac{1}{12} $, $ x_0 = -4 $, and $ y_0 = \frac{1}{6} $:
\[
y - \frac{1}{6} = \frac{1}{12}(x + 4)
\]
Simplifying this equation:
\[
y = \frac{1}{12}x + \frac{1}{3} + \frac{1}{6}
\]
\[
y = \frac{1}{12}x + \frac{1}{2}
\]

Thus, the equation of the tangent line is:
\[
y = \frac{1}{12}x + \frac{1}{2}
\]

Would you like further details or explanations on any part of the solution?

Here are 5 related questions to explore:

1. How do you find the derivative of a more complex rational function?
2. What is the geometric interpretation of a tangent line to a curve?
3. How would the tangent line change if the point of tangency was $ x = 4 $ instead of $ x = -4 $?
4. Can you find the equation of the normal line at $ x = -4 $?
5. What happens to the tangent line equation if the function is shifted vertically?

**Tip**: When differentiating functions like $ \frac{8}{3x^2} $, it helps to rewrite them in power form to apply the power rule easily.

This problem demonstrates how to find the equation of the tangent line to the function f(x) = 8/3x^2 at x = -4 using the power rule of differentiation. Learn how to calculate the derivative, determine the slope, and write the tangent line equation. Suitable for high school calculus students.

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