Partial Differentiation without chain rule in Euler Lagrange Equations

Question

The Euler-Lagrange equations for a bob attached to a spring are $$ \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial v} = \frac{\partial L}{\partial x} $$ But $v$ is a function of $x$. Is it or is it not. Because normal thinking says that $x$ is a function of $t$ and $v$ is a function of $t$ but it is not necessary that $v$ be a function of $x$. But mathematically
\begin{align} x&=f(t) \\ t&=f^{-1}(x) \\ v &= g(t) \\ &=g(f^{-1} (x)).\end{align} So chain rule should be applied in the Euler Lagrange equations. Then why is it not? Is it because on applying the chain rule on $f^{-1}$ and on applying the chain rule on both the sides of the equation we will get the same result. If not then what is the reason for not applying the chain rule? Or is it that $v$ is not a function of $x$ but then $v$ is related to $x$. Where am I going wrong in a such a basic thing is puzzling me a lot.