Partial Derivatives With N-Variables

• TranscendArcu
In summary, the conversation discusses finding the partial derivative of the geometric mean F(x_1,x_2,...,x_i,...,x_n) with respect to an arbitrary variable x_i. The solution is (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). The conversation then moves on to discussing maximizing F while constrained by G(x_1,x_2,...,x_n) = x_1 + x_2 + ... + x_n = c. The next step is to take the partial derivative of G with respect to x_i,

Homework Statement

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?

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)?

You got it!

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?

1. What is a partial derivative?

A partial derivative is a mathematical concept used in calculus to measure the rate of change of a function with respect to one of its variables, while holding all other variables constant. It is essentially the slope of the function in a specific direction.

2. How is a partial derivative denoted?

A partial derivative is denoted by the symbol ∂ (the partial derivative symbol) followed by the function name, and then the variable with respect to which the derivative is being taken. For example, the partial derivative of a function f(x,y) with respect to x would be written as ∂f/∂x.

3. What is the difference between a partial derivative and a regular derivative?

A regular derivative measures the rate of change of a function with respect to a single variable, while a partial derivative measures the rate of change of a function with respect to one variable while holding all other variables constant. Essentially, a partial derivative takes into account the influence of all other variables on the function, while a regular derivative does not.

4. How do you calculate a partial derivative?

To calculate a partial derivative, you treat all other variables in the function as constants and differentiate the function with respect to the variable in question. You can use the standard rules of differentiation, such as the power rule and product rule, to solve for the partial derivative.

5. What are some practical applications of partial derivatives?

Partial derivatives have many practical applications in fields such as physics, economics, and engineering. They are used to optimize functions and solve optimization problems, calculate marginal rates of change, and determine the sensitivity of a system to different variables. They are also used in the development of mathematical models to describe real-world phenomena.