https://file.aiquickdraw.com/mathbot/file/1728494773544uovwacqz.jpg

Let's solve the problem based on the provided image.

Given function:
\[ f(x) = \frac{-x}{1 + x^2} \]

### (a) Find the equation of the tangent line to this curve at the point (3, 0.30).

1. **Step 1: Find the derivative \( f'(x) \)**:
   \[
   f'(x) = \frac{d}{dx} \left( \frac{-x}{1 + x^2} \right)
   \]
   Using the quotient rule:
   \[
   f'(x) = \frac{(1 + x^2)(-1) - (-x)(2x)}{(1 + x^2)^2} = \frac{-1 - x^2 + 2x^2}{(1 + x^2)^2} = \frac{x^2 - 1}{(1 + x^2)^2}
   \]
   
2. **Step 2: Evaluate the slope of the tangent line at \( x = 3 \)**:
   \[
   f'(3) = \frac{3^2 - 1}{(1 + 3^2)^2} = \frac{9 - 1}{(1 + 9)^2} = \frac{8}{100} = 0.08
   \]
   So, the slope \( m = 0.08 \).

3. **Step 3: Use the point-slope form of the equation of a line**:
   The tangent line passes through the point \( (3, 0.30) \) and has slope \( m = 0.08 \).
   The point-slope form is:
   \[
   y - y_1 = m(x - x_1)
   \]
   Substituting \( x_1 = 3 \), \( y_1 = 0.30 \), and \( m = 0.08 \):
   \[
   y - 0.30 = 0.08(x - 3)
   \]
   Simplifying:
   \[
   y = 0.08x - 0.24 + 0.30
   \]
   \[
   y = 0.08x + 0.06
   \]
   Thus, the equation of the tangent line is:
   \[
   y = 0.08x + 0.06
   \]

### (b) The correct graph should show the curve \( f(x) \) and the tangent line \( y = 0.08x + 0.06 \) passing through the point (3, 0.30). 

Among the options shown in the image, **the second graph (middle)** correctly represents the tangent line and curve.

Let me know if you need further details or have any questions!

Here are 5 relative questions:
1. How do you apply the quotient rule in derivative calculations?
2. Why do we use the point-slope form to find the tangent line equation?
3. What is the significance of the slope of a tangent line at a given point?
4. Can you explain how to interpret the derivative geometrically?
5. How would the tangent line equation change if the point of tangency were different?

**Tip:** Always simplify derivatives before substituting values to make the calculation easier!

The graph of the function f(x) = -x / (1 + x^2) is called a serpentine. Find an equation of the tangent line to this curve at the point (3, 0.30).

Learn how to find the equation of the tangent line to the curve f(x) = -x / (1 + x^2) at the point (3, 0.30). This problem involves calculus concepts such as the derivative using the quotient rule and applying the point-slope form to determine the tangent line.

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