f(t) = ( 4 cos(t) - 4 cos(4t) , 4 sin (t) - 4 sin(4t) )

So f : R ⇒ R²

That function is differentiable at x0 if:

where Df(x0) is the matrix of partial derivatives. In this case:

Df = ( 16sin(4t) - 4sin(t) , 4cos(t) - 16cos(4t) )

Since there is only one variable in this function, there are no partials.

Try using that definition to see if it works.