I have to study the differentiability of the following curve
$$\begin{array}{cccc} C: & [0, 2\pi ]& \longrightarrow & \mathbb{C} \\
& t & \longmapsto & r(t+i-ie^{-ti}) \end{array}$$ at the points $t=0$ and $t=2\pi$.
Differentiating we obtain that
$$\begin{array}{cccc} C': & \mathbb{R} & \longrightarrow & \mathbb{C} \\ & t & \longmapsto &r\left( 1-e^{-ti}\right) \end{array}$$
My question is the following: is it enough to say that since
$$C^\prime (0)=0 $$ $$ C^\prime (2\pi)=0$$
the tangent vector is not well-defined, therefore the curve is not differentiable?
If not, could anyone point me how to do it properly?