This work focuses on the asymptotic behavior of the density in small time of a stochastic differential equation driven by an $\alpha$-stable process with index $\alpha \in (0,2)$. We assume that the process depends on a parameter $\beta=(\theta,\sigma)^T$ and we study the sensitivity of the density with respect to this parameter. This extends the results of Emmanuelle Clément and Arnaud Gloter which was restricted to the index $\alpha \in (1,2)$ and considered only the sensitivity with respect to the drift coefficient. By using Malliavin calculus, we obtain the representation of the density and its derivative as an expectation and a conditional expectation. This permits to analyze the asymptotic behavior in small time of the density, using the time rescaling property of the stable process.