variational principle - Independent Variables of a Lagrangian

Let us consider a particle in one spatial dimension $x$ and one temporal dimension $t$. Its Lagrangian $L$ is given by

$$\begin{eqnarray*} L &=& T- V \\ &=& \frac{1}{2} m\dot{x}^2 - V(x) \\ &=& L(x, \dot{x}) \end{eqnarray*}$$

But if I write $L = \frac{1}{2} m [\frac{d}{dt}({x})]^2 - V(x)$, then it seems to me that $L$ depends only on $x$.

So, my question is why it is said that $L$ is a function of $x$ and its derivative $\dot{x}$ ?