Derivation of the Takeoff equation
We want to reduce (or simplify) equation 2, to get get dV/dt by itself, so we multiply both sides by g/W.  On the right hand side, this reduces to dV/dt.
On the left hand side it reduces to what is given in equation 3, but let me show you how:
 1. First term is T x g/W
 2. Second term is -D x g/W
 3. Third term is -u x g/W (W - L).  By the distributive property, we have -[u x g/W x W] + [u g/W x L]. The first part of this term simplifies to u x g.

Finally, I want to group like terms together.


The first term combined with the first part of the third term give me: T x g/W - ug = g(T/w - u).
The second term combined with the second part of the third term give me: -D x g/W + u g/W L = g/W(u L - D).


This yields equation 3: g(T/w - u) + g/W(u L - D) = dV/dt.

Finally, recall that L = 1/2 p S V2 Cl, with a similar equation for Drag, and we can substitute these directly into L and D to get equation 4.

S is the the wing planform area. Other sources use A, like this one: NASA - Lift equation.