Why isn't the product rule used to derive x = 2sin(t)

Question

Since there is a multiplication occurring shouldn't the product rule be used? Or is the answer still dx/dt = 2cos(t) if the product rule is used? When I used the product rule I got 2cos(t) + sin(t).

Answer 1

Since the derivative is a linear operator, we can pull the constant out and differentiate $sin(t)$. However if you do apply the product rule:$$ \frac{dx}{dt}=2\cos(t)+(0)\sin(t)=2\cos(t)$$ The zero is due to the derivative of a constant being $0$. It looks like you messed up and took the derivative of $2$ to be $1$ instead of $0$.