Taking a time derivative of a function of 3 variables. I have a function of $3$ variables which are all functions of $t$.
$$x = \frac{v_1t-y}{\sqrt{(v_2/\dot{x})^2 -1}} \tag 1 $$ 
In the equation $v_1,v_2$ are constant and $x$ and $y$ are both function of $t$ (also $\dot{x}$ is $\frac{dx}{dt}$). I am trying to differentiate $(1)$ with respect to $t$, but I am not sure how to do this as there are three variables and an $\dot{x}$ already. I tried holding $x,y$ constant but that instinctively does not make sense as these change with respect to $t$.

Hint:
If $y(t)$ and $\dot{x}(t)$ are two functions independent from $x(t)$
 you can use quotient rule and chain rule.
If $\dot{x}=\dfrac{dx}{dt}$, as I suppose, you have not a function but a differential equation that you can manipulate formally solving for $\dot{x}$ and find:
$$
\dot{x}=\dfrac{dx}{dt}=F(t,x(t),y(t))=\dfrac{v_2x}{\sqrt{(v_1t-y)+x^2}}
$$
But it seems not simple to solve.
