# What is the derivative of the kinetic energy function?

Jul 9, 2018

It gives us the momentum equation respect to velocity...

#### Explanation:

The function or equation for kinetic energy is:

$\boldsymbol{K E} = \frac{1}{2} m {v}^{2}$

Taking the derivative respect to velocity $\left(v\right)$ we get:

$\frac{d}{\mathrm{dv}} \left(\frac{1}{2} m {v}^{2}\right)$

Take the constants out to get:

$= \frac{1}{2} m \cdot \frac{d}{\mathrm{dv}} \left({v}^{2}\right)$

Now use the power rule, which states that $\frac{d}{\mathrm{dx}} \left({x}^{n}\right) = n {x}^{n - 1}$ to get:

$= \frac{1}{2} m \cdot 2 v$

Simplify to get:

$= m v$

If you learn physics, you should clearly see that this is the equation for momentum, and states that:

$p = m v$