Answer
For $\displaystyle \frac{\partial K}{\partial m}$, m is the variable, all other variables are treated as constants.
$\displaystyle \frac{\partial K}{\partial m}=\frac{\partial}{\partial m}[\frac{1}{2}mv^{2}]=\frac{1}{2}v^{2}$
For $\displaystyle \frac{\partial K}{\partial v},\ v$ is the variable, all other variables are treated as constants.
$\displaystyle \frac{\partial K}{\partial v}=\frac{\partial}{\partial v}[\frac{1}{2}mv^{2}]=mv$
$\displaystyle \frac{\partial^{2}K}{\partial v^{2}}=\frac{\partial}{\partial v}[mv]=m$
$\displaystyle \frac{\partial K}{\partial m}\cdot\frac{\partial^{2}K}{\partial v^{2}}=\frac{1}{2}v^{2}m=K,$
which is what was needed to be shown.
Work Step by Step
All steps are included in the answer.