# Partial derivative help for sigma and pi notation in Lagrange maximization

For a high school multivariable calc class I am writing an essay on an economic utility function and its maximisation using the Lagrange multipliers.

The formula is $output = \prod_i^n x_i^{\sigma_i}$, where n is the number of goods, x is a specific good and $\sigma$ is a multiplier. The constraint needed for the lagrange multiplier method is C > $\sum_i^n x_i*p_i$ where p is the price and C is the maximum cost of a company.

So essentially you have: $$f(x_i) = \prod_i^n x_i^{\sigma_i} - {\lambda}(C-\sum_i^n x_i*p_i)$$

Now my question is how to go on. Which partial derivative should I take, and also is that possible with the pi and sigma notation?

When differentiating a monomial with respect to one of the variables, the other variables are to be held constant. Therefore, $$\frac{\partial}{\partial x_j} \left(x_1^{\sigma_1} x_2^{\sigma_2} \dots x_n^{\sigma_n}\right) = x_1^{\sigma_1} x_2^{\sigma_2} \cdots x_{j-1}^{\sigma_{j-1}}\sigma_j x_j^{\sigma_j-1} x_{j+1}^{\sigma_{j+1}}\dots x_n^{\sigma_n}$$ As long as $x_j \neq 0$ you could write it as $$\frac{\partial}{\partial x_j}\prod_{i=1}^n x_i^{\sigma_i} = \frac{\sigma_j}{x_j} \prod_{i=1}^n x_i^{\sigma_i}$$
When differentiating a linear combination with respect to one of the variables, again the other variables are to be held constant. Therefore, $$\frac{\partial}{\partial x_j}\left(p_1 x_1 + p_2 x_2 + \dots + p_n x_n\right) = \frac{\partial}{\partial x_j} \sum_{i=1}^n p_i x_i = p_j$$