Find matrix of derivative

Question

A robot traverses a path γ : [0,1] -> R^2 given by γ(t)=(t sin⁡((π/2)t)) (sorry this is a 2x1 matrix).

First, explain why the robot remains on the plate. Then write down the matrix γ' of the derivative of γ at general t ∈ (0,1).

Not really sure where to start?

Answer 1

I think you mean $$ \gamma(t)=(t,\sin (\pi/2 t)) $$ in which case you find the derivative of the curve component-wise $$ \gamma'(t)=(1,\pi/2\cos (\pi/2 t)) $$