Let $f:[0,1]\to\Bbb{R}$ a continuous function which is differentiable in $(0,1)$, but is not in $0$ and $1$ I need to draw a graphic which the function is differentiable in whole $(0,1)$ but is not in $0$ and $1$.
I just can imagine the function $f(x)=cotg(\frac{x}{\pi})$ but this $f$ is not well-defined in $[0,1]$, despiste being not differentiable at $0$ and $1$.

You've already seen
$$
f_{1}(x) = \sqrt{x - x^{2}},
$$
whose graph is the top half of a circle over $[0, 1]$; other examples include suitable scalings of $\arccos$ or $\arcsin$, such as
$$
f_{2}(x) = \arcsin\bigr(\tfrac{1}{2}(1 + x)\bigr),
$$
which again has vertical tangents at the endpoints.
To get more spectacular failure, try something like
$$
f_{3}(x) = f_{1}(x) \sin\left(\frac{1}{f_{1}(x)^{n}}\right),\quad n \geq 1,
$$
extended by continuity to be $0$ at the endpoints. Here, the derivative oscillates (and is unbounded) near each endpoint, so the endpoint derivatives do not exist even if you allow $\pm\infty$ as values.

