Tutorial Exercise Find the derivative of the function. \[ f(t)=6 t \sin (\pi t) \] Step 1 We note that the given function \( f(t)=6 t \sin (\pi t) \) is the product of two differentiable functions of the form \( f(t)=g(t) h(t) \) where \( g(t)=6 t \) and \( h(t)=\sin (\pi t) \). However, before we can apply the product rule we must first find \( h^{\prime}(t) \). Doing so requires the use of the chain rule because \( h(t)=\sin (\pi t) \) is a composite function with \( u=\pi t \) and \( h(u)=\sin (u) \). Furthermore, we recall that, in general, if \( y=\sin (u) \), where \( u \) is a differentiable function of \( t \), then, by the chain rule, we have the following. \[ \begin{array}{l}\frac{d y}{d t}=\frac{d y}{d u} \frac{d u}{d t}=\cos (u) \frac{d u}{d t} \\ \quad u=\pi t \\ \frac{d u}{d t}=\square \\ \text { So, we first find } \frac{d u}{d t} \\ \text { SUBMIT }\end{array} \]