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The problem requires finding \( \frac{dy}{dt} \) using implicit differentiation, given the equation \( y = x^2 \) and \( \frac{dx}{dt} = 6 \), with \( x = 3 \).

### Steps:

1. **Differentiate implicitly** with respect to \( t \):
   \[
   \frac{d}{dt}(y) = \frac{d}{dt}(x^2)
   \]
   Applying the chain rule:
   \[
   \frac{dy}{dt} = 2x \frac{dx}{dt}
   \]

2. **Substitute the known values**:
   - \( x = 3 \)
   - \( \frac{dx}{dt} = 6 \)

   So,
   \[
   \frac{dy}{dt} = 2(3)(6) = 36
   \]

Thus, the value of \( \frac{dy}{dt} \) is **36**.

Would you like more details on implicit differentiation or anything else? Here are five related questions to expand your understanding:

1. How does implicit differentiation differ from explicit differentiation?
2. Can implicit differentiation be applied when both \( x \) and \( y \) are functions of \( t \)?
3. What is the chain rule, and how does it apply in this case?
4. How would the solution change if \( \frac{dx}{dt} \) were a different constant?
5. How can you solve similar problems when \( y \) is a more complex function of \( x \)?

**Tip**: Always apply the chain rule when differentiating expressions where multiple variables depend on another variable, like \( t \).

Derivación implícita respecto a t: Si y = x^2 y dx/dt = 6, entonces, cuando x = 3, encontrar dy/dt.

Learn how to apply implicit differentiation to find dy/dt given y = x^2 and dx/dt = 6, with x = 3. This problem focuses on the chain rule and basic calculus principles, suitable for high school and early college students.

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