Two different definitions for Lie Algebras for closed subgroup of $GL_n(\mathbb R)$

Question

Let $G$ be a closed subgroup of $GL_n(\mathbb R)$.

There are two definitions for $\mathrm{Lie}(G)$

1. $\mathrm{Lie}(G) = \{ \gamma'(0) : \gamma : (-\epsilon, \epsilon) \rightarrow G \text{ is differentiable with } γ(0) = I\}$.

2. $\mathrm{Lie}(G) = \{ A \in M_n( \mathbb R ) : e^{At} \in G, \forall t \in \mathbb R \}$.

Why are these two definitions equivalent?

PS. I have no background in differential geometry.