To find the next (third) derivative of \( y \) with respect to \( x \), denoted as \(\frac{d^3y}{dx^3}\), we need to take the derivative of \(\frac{d^2y}{dx^2}\) with respect to \( x \).

Recall that:
\[ \frac{d^2y}{dx^2} = -\frac{1}{32t^3} \]

We need to use the chain rule again to differentiate \(\frac{d^2y}{dx^2}\) with respect to \( x \):

\[ \frac{d}{dx}\left( \frac{d^2y}{dx^2} \right) = \frac{\frac{d}{dt} \left( \frac{d^2y}{dx^2} \right)}{\frac{dx}{dt}} \]

First, compute \(\frac{d}{dt} \left( \frac{d^2y}{dx^2} \right)\):

\[ \frac{d}{dt} \left( -\frac{1}{32t^3} \right) = -\frac{d}{dt} \left( \frac{1}{32t^3} \right) = -\frac{1}{32} \cdot \frac{d}{dt} \left( t^{-3} \right) = -\frac{1}{32} \cdot (-3t^{-4}) = \frac{3}{32t^4} \]

Next, divide this by \(\frac{dx}{dt} = 2t\):

\[ \frac{d^3y}{dx^3} = \frac{\frac{d}{dt} \left( \frac{d^2y}{dx^2} \right)}{\frac{dx}{dt}} = \frac{\frac{3}{32t^4}}{2t} = \frac{3}{32t^4} \cdot \frac{1}{2t} = \frac{3}{64t^5} \]

Now, evaluate at \( t = 2 \):

\[ \frac{d^3y}{dx^3} \Bigg|_{t=2} = \frac{3}{64(2)^5} = \frac{3}{64 \cdot 32} = \frac{3}{2048} = \frac{3}{2048} \]

Thus, the third derivative of \( y \) with respect to \( x \) at \( t = 2 \) is:

\[ \frac{d^3y}{dx^3} = \frac{3}{2048} \]

Would you like further details or have any questions? Here are some questions you might consider:

1. How do you find higher-order derivatives for parametric equations?
2. Can you explain the process of using the chain rule repeatedly for higher-order derivatives?
3. What are some applications of third derivatives in physics and engineering?
4. How do the first, second, and third derivatives relate to the shape of a graph?
5. Can you provide examples of other parametric equations and their higher-order derivatives?

**Tip:** When finding higher-order derivatives for parametric equations, carefully apply the chain rule step-by-step and simplify at each stage to avoid errors.

Learn how to find the third derivative of y with respect to x using the chain rule and parametric equations. This solution covers step-by-step differentiation processes and applications in higher-order calculus. Suitable for advanced calculus students and enthusiasts.

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