Find the derivative of the function.
$ f(t) = t \sin \pi t $
$\pi t \cos \pi t+\sin \pi t$
here's our function and we're going to find its derivative and this is actually a product where T is a factor and sign of pi times t is a factor, so we're going to use the product rule to differentiate. So we have The derivative is the first tee times the derivative of the second eso. In finding the derivative of the second, we're going to need to use the chain rule and the outer function. There is the sine function and the inter function there is pie T. So the derivative of the outside would be the co sign so co sign of pi t times the derivative of the inside the derivative of pi ti would be pie. So what we have so far in the process of using the product rule is the first times the derivative of the second we still need. Plus, the second time's a derivative of the first. So the second is the sign of pi t times the derivative of the first. The derivative of tea would be one. Okay, we have our derivative. Now we're just going to simplify it. So let's go ahead and put the pie and the tea together in that first term. So we have pie times t times a co sign of pi t And then the second term is just plus the sign of pi t.