Let $ \Omega = \{ x \in \mathbb{R} : | x | \leq 1 \} $. Show that the function $ v(x) = |x|^{\alpha} $ belong to $ H^1(\Omega) $ if $ \alpha > 0 $

Question

Let $ \Omega = \{ x \in \mathbb{R} : | x | \leq 1 \} $. Show that the function $ v(x) = |x|^{\alpha} $ belong to $ H^1(\Omega) $ if $ \alpha > 0 $.

We know that:

$$ \mathbb{L}_2(\Omega) = \{ v: v \ \text{its defined in} \ \Omega \ \text{and} \ \int_{\Omega} v^2 dx < \alpha \} $$

and that

$$ H^1(\Omega) = \{ v \in \mathbb{L}_2 (\Omega) : \frac{\partial v}{ \partial x_i} \in \mathbb{L}_2(\Omega) , i = 1 , 2 , \ldots , d \} $$

where $ d \text{ comes from } R^d $

I did'nt understand how to solve this problem, i have tried numerically but my teacher says that im going in the wrong direction

Appreciate any help

thanks