Partial Derivatives With N-Variables

1. The problem statement, all variables and given/known data

Given F(x_1,x_2,...,x_i,...,x_n) = nth-root(x_1*x_2*...*x_i*...*x_n), how do I take the partial derivative with respect to x_i, where x_i is an arbitrary variable?

2. Relevant equations



3. The attempt at a solution

Would it just be:

(1/n)(x_1*x_2*...*x_i*...*x_n)^((1/n)-1)*(x_1*x_2*...*x_i-1*x_i+1*...*x_n)?

 

Alright, nice!

What I'd like to try to do with this problem is maximize F (the equation for the geometric mean) when it is constrained by G(x_1,x_2,...,x_n) = x_1 + x_2 + ... + x_n = c, where c is some constant. I can take the partial derivative of G with respect to x_i and get,

G_x_i = 1

But I don't really know if I am a) on the right track for this problem or b) how to proceed if I am on the right track. What should my next step be?