A rock falling under its on weight and resistance r=0.001mv^2
show newtons second law gives vdv/dx=g-0.001v^2
Am I missing something?? when did vdv/dx = a having cancelled through by m
You should know $\displaystyle a = \frac{dv}{dt} = \frac{d^2 x}{dt^2}$ and $\displaystyle v = \frac{dx}{dt}$.
Use the chain rule:
$\displaystyle {\color{red}a = \frac{dv}{dt} = \frac{dv}{dx} \cdot \frac{dx}{dt} = \frac{dv}{dx} \cdot v}$.
Using the chain rule again it follows that $\displaystyle a = \frac{d}{dx} \left[\frac{v^2}{2} \right]$.