To find the equation of the tangent line to the graph of the function $f(x) = \frac{\sqrt{x}}{3x + 3}$ at the point $(3, f(3))$, we need to follow these steps:

1. Calculate $f(3)$:
   Plug $x = 3$ into the function to find the y-coordinate of the point of tangency.

   $f(3) = \frac{\sqrt{3}}{3(3) + 3} = \frac{\sqrt{3}}{9 + 3} = \frac{\sqrt{3}}{12}$

   So, the point of tangency is $\left(3, \frac{\sqrt{3}}{12}\right)$.

2. Find the derivative $f'(x)$:
   The derivative of $f(x)$ gives the slope of the tangent line. To find $f'(x)$, we use the quotient rule:

   $f(x) = \frac{\sqrt{x}}{3x + 3}$

   The quotient rule states: $f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}$ where $g(x) = \sqrt{x}$ and $h(x) = 3x + 3$.

   First, find the derivatives: $g'(x) = \frac{1}{2\sqrt{x}}, \quad h'(x) = 3$

   Then, apply the quotient rule: $f'(x) = \frac{\left(\frac{1}{2\sqrt{x}}\right)(3x + 3) - \left(\sqrt{x}\right)(3)}{(3x + 3)^2}$

   Simplify this expression: $f'(x) = \frac{\frac{3x + 3}{2\sqrt{x}} - 3\sqrt{x}}{(3x + 3)^2}$

   Simplifying further: $f'(x) = \frac{\frac{3(x + 1) - 6x}{2\sqrt{x}}}{(3x + 3)^2} = \frac{\frac{3 - 3x}{2\sqrt{x}}}{(3x + 3)^2}$

3. Evaluate $f'(3)$:
   Substitute $x = 3$ into the derivative to find the slope at $x = 3$:

   $f'(3) = \frac{\frac{3(1 - 3)}{2\sqrt{3}}}{(3(3) + 3)^2} = \frac{\frac{3(-2)}{2\sqrt{3}}}{12^2} = \frac{-3\sqrt{3}}{72} = \frac{-\sqrt{3}}{24}$

   So, the slope of the tangent line at $x = 3$ is $\frac{-\sqrt{3}}{24}$.

4. Use the point-slope form of the line: The equation of the tangent line is given by:

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

   Here, $(x_1, y_1) = \left(3, \frac{\sqrt{3}}{12}\right)$ and $m = \frac{-\sqrt{3}}{24}$.

   Substituting these values: $y - \frac{\sqrt{3}}{12} = \frac{-\sqrt{3}}{24}(x - 3)$

   Simplifying: $y = \frac{-\sqrt{3}}{24}x + \frac{\sqrt{3}}{8} + \frac{\sqrt{3}}{12}$

   Combine the constant terms: $y = \frac{-\sqrt{3}}{24}x + \frac{\sqrt{3}}{24}$

So, the equation of the tangent line is:

$y = \frac{-\sqrt{3}}{24}(x - 1)$

Would you like further details or have any questions?

Related Questions:

1. How does the quotient rule apply to different types of functions?
2. What is the geometric interpretation of a derivative?
3. How can the slope of a tangent line help in understanding the behavior of a function?
4. What is the importance of finding the tangent line in real-world applications?
5. How would you find the tangent line for more complex functions?

Tip: When using the quotient rule, carefully keep track of each component function's derivative to avoid errors.