homework and exercises - Obtaining a general equation for velocity (in 2D projectile motion)

I'm trying to obtain a general equation for the instantaneous velocity of a projectile moving on a Cartesian plane.

I began with the equation for a projectile's trajectory (air resistance neglected):

$$y = x(tanθ) - \frac {gx^2}{(u^2)(cosθ)^2}$$

where $u$ is the projection velocity, and $θ$ is the projection angle.

I then sought to differentiate the above-mentioned equation with respect to time. This yielded:

$$y' = x'(tanθ) - \frac {2gxx'}{(u^2)(cosθ)^2}$$

Where $'$ stands for a differential with respect to time.

Now, re-writing the equation:

$$v_y = v_x(tanθ) - \frac {2gxv_x}{(u^2)(cosθ)^2}$$

Where $v_y$ and $v_x$ are the $y$ and $x$ components of instantaneous velocity.

My issue?

I can't seem to be able to get the last equation in terms of the variables $v_y$ and $v_x$ alone (I can't seem to eliminate the $x$).

My question:

Is it possible to obtain a general equation for instantaneous velocity with $v_y$ and $v_x$ as the only variables? If so, how do I go about it?