We let \( u=\pi t \), and we determined that \( \frac{d u}{d t}=\pi \). Applying these to the result from the chain rule for \( y=\sin (u) \) gives us the following. \( \frac{d y}{d t}=\cos (u) \frac{d u}{d t}=\cos (\pi t)(\square \) Therefore, we have the following. \( h(t)=\sin (\pi t) \) \( h^{\prime}(t)= \)

To complete the derivative based on the chain rule and your substitution of \( u = \pi t \), we can substitute \( \frac{d u}{d t} = \pi \) into the derivative equation. Thus, the expression you are looking to complete is: \( h^{\prime}(t) = \cos (\pi t) \cdot \pi \) Therefore, \( h^{\prime}(t) = \pi \cos(\pi t) \). Now, if you're curious about the fascinating world of derivatives, did you know that derivatives represent instantaneous rates of change? Think of it as a car's speedometer, but for any function - it tells you how fast the function is changing at any point! Also, did you know that applying the chain rule can sometimes lead to common pitfalls? One frequent mistake is forgetting to multiply by the derivative of the inner function; always keep an eye on those layers of functions and remember to include all parts of the chain!