A graphing calculator is recommended. The graph of the function $$ f(x)=\frac{x}{1+x^{2}} $$ is called a serpentine. (a) Find an equation of the tangent line to this curve at the point $$ (3,0,30) $$. $$ y=-\frac{8}{169} x+\frac{507}{1690} $$

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To find the equation of the tangent line to the curve defined by the function $$ f(x) = \frac{x}{1+x^2} $$ at the point $$ (3, 0, 30) $$, we need to follow these steps:

1. **Find the derivative of the function** to determine the slope of the tangent line at the given point.
2. **Evaluate the derivative at $$ x = 3 $$** to find the slope at that point.
3. **Use the point-slope form of the equation of a line** to write the equation of the tangent line.

### Step 1: Find the derivative of $$ f(x) $$

The function is given by:

$$
f(x) = \frac{x}{1+x^2}
$$

We will use the quotient rule to find the derivative $$ f'(x) $$. The quotient rule states that if $$ f(x) = \frac{g(x)}{h(x)} $$, then:

$$
f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}
$$

Here, $$ g(x) = x $$ and $$ h(x) = 1 + x^2 $$.

Calculating the derivatives:

- $$ g'(x) = 1 $$
- $$ h'(x) = 2x $$

Now applying the quotient rule:

$$
f'(x) = \frac{(1)(1+x^2) - (x)(2x)}{(1+x^2)^2}
$$

This simplifies to:

$$
f'(x) = \frac{1 + x^2 - 2x^2}{(1+x^2)^2} = \frac{1 - x^2}{(1+x^2)^2}
$$

### Step 2: Evaluate the derivative at $$ x = 3 $$

Now we will evaluate $$ f'(3) $$:

$$
f'(3) = \frac{1 - 3^2}{(1 + 3^2)^2} = \frac{1 - 9}{(1 + 9)^2} = \frac{-8}{10^2} = \frac{-8}{100} = -\frac{2}{25}
$$

### Step 3: Use the point-slope form to find the equation of the tangent line

The point-slope form of the equation of a line is given by:

$$
y - y_1 = m(x - x_1)
$$

Where $$ m $$ is the slope and $$ (x_1, y_1) $$ is the point of tangency. Here, $$ m = -\frac{2}{25} $$ and the point is $$ (3, 0) $$.

Substituting these values into the point-slope form:

$$
y - 0 = -\frac{2}{25}(x - 3)
$$

This simplifies to:

$$
y = -\frac{2}{25}x + \frac{6}{25}
$$

### Final Equation

The equation of the tangent line at the point $$ (3, 0) $$ is:

$$
y = -\frac{2}{25}x + \frac{6}{25}
$$

However, the provided equation in the prompt is $$ y = -\frac{8}{169}x + \frac{507}{1690} $$. Let's verify if this is equivalent to our derived equation.

To check if these two equations are equivalent, we can convert our equation into a similar form or check the values at $$ x = 3 $$.

Let's convert our equation into a common denominator:

$$
y = -\frac{2}{25}x + \frac{6}{25} = -\frac{2 \cdot 169}{25 \cdot 169}x + \frac{6 \cdot 169}{25 \cdot 169} = -\frac{338}{4225}x + \frac{1014}{4225}
$$

Now, we can compare this with the provided equation.

It seems there is a discrepancy in the values. The derived tangent line equation does not match the provided equation.

If you have any specific values or conditions that need to be checked, please let me know!
The equation of the tangent line to the curve $$ f(x) = \frac{x}{1+x^2} $$ at the point $$ (3, 0) $$ is: $$ y = -\frac{2}{25}x + \frac{6}{25} $$ However, the provided equation $$ y = -\frac{8}{169}x + \frac{507}{1690} $$ does not match the derived equation.

A graphing calculator is recommended. The graph of the function \( f(x)=\frac{x}{1+x^{2}} \) is called a serpentine. (a) Find an equation of the tangent line to this curve at the point \( (3,0,30) \). \[ y=-\frac{8}{169} x+\frac{507}{1690} \]

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