A force f[x] imparts an acceleration a[x] to an object of constant mass m moving (with no friction) along the x-axis from x = a to x = b.

Physicists have a notion of the work done by this force while it moves the object from x = a to x = b. They measure work with an integral:

Work = Integrate[m a[x], {x, a, b}].

Why is

a[x] = v[x]v'[x]

where v[x] is velocity at position x for a x b?

Tip - Remember that velocity is the derivative of position with respect to time, and acceleration is the derivative of velocity with respect to time. Use the chain rule: