Question

What is the derivative of kinetic energy with respect to velocity?

Answer 1

It's the linear momentum `p=mv`.

The kinetic energy of a particle is defined as `K=1/2 mv^2`.

It's derivative with respect to the the velocity `v` is:

`(dK)/(dv)=d/(dv)[1/2 mv^2]`

Since the mass `m` does not depend on the velocity and the factor `1/2` is constant, the linear property of the derivative gives us:

`d/(dv)[1/2 mv^2]=1/2 m d/(dv) [v^2]`

Knowing the derivative of a power function `d/(dx)[x^n]=n x^(n-1)` gives us the result:

`(dK)/(dv)=1/2 m 2 v=mv=p`

This answer is valid if we consider the classical case. Taking into account relativistic effects gives us the same result, but the derivation is more complicated.