Intuition behind adjoint map

Question

Let $G$ be a Lie group, and let $g,h\in G$. Suppose we have the map $$\Lambda_g:G\to G$$ such that $$h\to ghg^{-1}$$ This induces a map $\mathfrak{ad}_g$ on the tangent spaces such that $$\mathfrak{dg}_g: X\to \frac{\partial}{\partial t}ge^{tX}g^{-1}$$ Why do we consider $e^{tX}$ instead of $X$ directly?