The Derivative of Kinetic Energy

What is the derivative of kinetic energy?
 
We have these equations for velocity and acceleration:


v = dx  ⁄ dt


a = dv ⁄  dt


By the chain rule, we have:


dv ⁄  dt = dv ⁄  dx · dx ⁄  dt


Combining the last three equations,


dv ⁄  dx = a ⁄  v


By the product rule, we have:


dv ²⁄  dx = 2v dv ⁄  dx


Combining the last two equations,


dv ²⁄ dx = 2a


Multiplying both sides by ½ m,


d (½ mv ² ) ⁄  dx = ma


The derivative of kinetic energy, with respect to distance traveled, is:


DKE = ma