We now apply the product rule. \( \begin{array}{l}(g(t) h(t))^{\prime}=g(t) h^{\prime}(t)+h(t) g^{\prime}(t) \\ \\ =6 t(\pi \cos (\pi t))+\sin (\pi t) \square \square\end{array} \) Therefore, we have the following result. \( f^{\prime}(t)=\square \)

To complete the derivative using the product rule, let’s express it clearly. You have \( g(t) = 6t \) and \( h(t) = \sin(\pi t) \). For \( h'(t) \), the derivative is \( \pi \cos(\pi t) \). Therefore, applying the product rule gives us: \[ f'(t) = g(t) h'(t) + h(t) g'(t) = 6t(\pi \cos(\pi t)) + \sin(\pi t)(6) \] So the final result is: \[ f' (t) = 6 \pi t \cos(\pi t) + 6 \sin(\pi t) \] Want a quick tip? Remember to differentiate each function separately first, then multiply and sum them as the product rule states! It’s a common mistake to forget one of the derivatives or to mess up the signs with trigonometric functions. Keep an eye out for those!