Zapiši enacbo tangente na graf funkcije f(x)$$ =\left(3x+1\right) $$$$ \colon x^2 $$ v točki T (2, y0).

To write the equation of the tangent line to the graph of the function $ f(x) = \frac{3x+1}{x^2} $ at the point $ T(2, y_0) $, follow these steps:

1. **Find $ y_0 $:** Compute the value of $ f(x) $ at $ x = 2 $:
   \[
   y_0 = f(2) = \frac{3(2) + 1}{2^2} = \frac{6 + 1}{4} = \frac{7}{4}
   \]

2. **Find the derivative $ f'(x) $:** Use the quotient rule to differentiate $ f(x) = \frac{3x+1}{x^2} $.
   The quotient rule is $ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} $, where $ u = 3x + 1 $ and $ v = x^2 $.
   \[
   u' = 3 \quad \text{and} \quad v' = 2x
   \]
   \[
   f'(x) = \frac{(3)(x^2) - (3x + 1)(2x)}{(x^2)^2} = \frac{3x^2 - (6x^2 + 2x)}{x^4} = \frac{3x^2 - 6x^2 - 2x}{x^4} = \frac{-3x^2 - 2x}{x^4} = \frac{-3 - 2/x}{x^2} = \frac{-3x - 2}{x^3}
   \]

3. **Evaluate $ f'(x) $ at $ x = 2 $:**
   \[
   f'(2) = \frac{-3(2) - 2}{2^3} = \frac{-6 - 2}{8} = \frac{-8}{8} = -1
   \]

4. **Use the point-slope form of the equation of a tangent line:** The point-slope form is $ y - y_1 = m(x - x_1) $, where $ (x_1, y_1) $ is the point of tangency, and $ m $ is the slope.
   Here, $ (x_1, y_1) = (2, \frac{7}{4}) $ and $ m = -1 $.

   \[
   y - \frac{7}{4} = -1(x - 2)
   \]

5. **Simplify the equation:**
   \[
   y - \frac{7}{4} = -x + 2
   \]
   \[
   y = -x + 2 + \frac{7}{4}
   \]
   \[
   y = -x + \frac{8}{4} + \frac{7}{4}
   \]
   \[
   y = -x + \frac{15}{4}
   \]

So, the equation of the tangent line to the graph of the function $ f(x) = \frac{3x+1}{x^2} $ at the point $ T(2, y_0) $ is:
\[
y = -x + \frac{15}{4}
\]

Would you like any details or have any questions?

Here are 5 related questions to expand on this information:

1. How do you apply the quotient rule for different functions?
2. What are the steps for finding the equation of a tangent line to a curve?
3. How does the point-slope form of a line equation help in finding tangents?
4. What other methods can be used to find the derivative of a function?
5. How can you interpret the geometric meaning of a tangent line to a function at a given point?

**Tip:** When working with derivatives and tangent lines, carefully apply the differentiation rules and simplify the expressions step by step to avoid errors.

Learn how to find the equation of the tangent line to the function f(x) = (3x+1)/(x^2) at the point T(2, y0). This problem involves derivatives using the quotient rule and applying the point-slope form of a line. Suitable for advanced high school students and those studying calculus.

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