general relativity - How does Newton's 2nd law correspond to GR in the weak field limit?

I can only perform the demonstration from the much simpler $E = mc^2$.

Take as given the Einstein field equation:

$G_{\mu\nu} = 8 \pi \, T_{\mu\nu}$

... can it be proved that Newton's formulation of gravitational and mechanical force (e.g. $F = ma$) corresponds to Einstein's in the limit when masses are small and speeds are relatively slow?