Finding the derivative of the vector function $ \mathbf{r}(t) = \langle t \sin(t),t^{2},t \cos(2t) \rangle $.

Question

I have this problem in my book, and the answer (lower half). I'm completely lost as to why there are extra trig functions being added and subtracted from. I thought I was just taking the normal derivative of each part but... Can someone explain this?

Answer 1

Be reminded that Product rule applies when you differentiate the product of $2$ functions:

$$\frac d{dx}[f(x)g(x)]=f(x)g'(x)+f'(x)g(x)$$

For instance, in you context,

$$\frac {d(t\sin t)}{dt}=t(\sin t)'+(t)'\sin t=t\cos t+\sin t$$

Answer 2

They are no added and subtracted functions, just remember to apply correctly de derivate of the product in the first and third entry.