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 $t$.