Local complete intersection scheme, conormal sheaves and differentials

Question

Let $X$ be a smooth projective variety over $\mathbb{C}$ and $Z \subset X$ be a local complete intersection subscheme in $X$. Denote by $I_Z$ the ideal sheaf of $Z$ in $X$ and $\Omega^1_X$ the sheaf of Kähler differentials. Then, is it true that the map from $I_Z/I_Z^2 \to \Omega^1_X \otimes \mathcal{O}_Z$ induced by the differential map, injective? If not, without assuming $Z$ to be smooth is there any condition, under which this map can be injective, for example if $Z$ is a divisor?