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