How can this be interpreted physically?

Question

Mathematically, the magnitude of the momentum of an object is equal to the derivative of its kinetic energy with respect to its speed. What physical implications does this have?

Answer 1

The derivative of Potential energy with respect to position is Force. The derivative of momentum with respect to time is Force.

As you say the derivative of kinetic energy with respect to velocity is momentum. Velocity is the derivative of position with respect to time. So the derivative of kinetic energy with respect to the derivative of position with respect to time is equal to mass times the derivative of position with respect to time.

You are right, there is definitely something here...