Why is velocity the derivative of energy over momentum?

In quantum mechanics, the energy is the frequency of the wave, and the momentum the wavenumber (up to factors of hbar, which you can set to one using natural units). read more

The derivative of the energy with respect to the wavenumber is the group velocity of a wavepacket, so this is the velocity of a classical particle, which is what happens in the limit where the wavepacket size and the wavelength, and every other wave scale, goes to zero. read more

Kinetic energy is the integral of momentum with respect to velocity: $$\int mv \cdot dv = \frac{1}{2}mv^2$$ The fact that each of these are integrals/derivatives of the other probably hints at some deeper connection. read more

To understand why kinetic energy is the integral of momentum with respect to vleocity (or equivalently, that momentum = the derivative of kinetic energy with respect to velocity), you have to know a little about Lagrangian mechanics, a reformulation of Newtonian mechanics that is much more elegant and powerful. read more