The utility function of a consumer is $u= (x_1^{1/3})\cdot (x_2^{2/3}) $ and he seeks to minimize his budget. In this case $x_1$ and $x_2$ are the goods and $p_1$ and $p_2$ are their prices respectively. In this minimization problem I set it up in the following manner:
Minimized: $p_1\cdot x_1 + p_2 \cdot x_2$ for $u =(x_1^{1/3})\cdot (x_2^{2/3}) $
Then I used the Lagrangian to find the values of $x_1,x_2$ and the multiplier $\lambda$. They turned out to be $x_1 = u (p_2/2p_1)^{1/3}, x_2 = u (2p_1/p_2)^{1/3}$ and the Lagrange multiplier being $3u (p_1\cdot x^{2/3})\cdot (p_2/2p_1)^{1/3}.$ I am not sure if I calcuated the Lagrange multiplier correctly. However, my main query is that I want to find the value function V and then evaluate $dV/dp_i$ where $i=1,2$. I understand what the value function means in brief but I am having difficulty understanding how to work out this portion from the aforementioned things I've calculated.