Answer
For $\displaystyle \frac{\partial P}{\partial L}$, L is the variable, all other variables are treated as constants.
$\displaystyle \frac{\partial P}{\partial L}=\alpha bL^{\alpha-1}K^{\beta}$
For $\displaystyle \frac{\partial P}{\partial K}$, K is the variable, all other variables are treated as constants.
$\displaystyle \frac{\partial P}{\partial K}=\beta bL^{\alpha}K^{\beta-1}$
LHS=$L\displaystyle \frac{\partial P}{\partial L}+K\frac{\partial P}{\partial K}$
$= L(\alpha bL^{\alpha-1}K^{\beta})+K(\beta bL^{\alpha}K^{\beta-1})$
$=\alpha bL^{\alpha}K^{\beta}+\beta bL^{\alpha}K^{\beta}$
$=(\alpha+\beta)bL^{\alpha}K^{\beta}$
$=(\alpha+\beta)P$ =RHS
Work Step by Step
All steps shown in the answer.