Math Problem Statement

Given the equations: x = y^2 + z^2, y = x + z^2, z = n * y, and w = x + y + z, find the partial derivative ∂w/∂(xyz).

Solution

The provided image shows a mathematical expression with the following equations and expression:

  1. x=y2+z2x = y^2 + z^2
  2. y=x+z2y = x + z^2
  3. z=nyz = n \cdot y
  4. w=x+y+zw = x + y + z

The goal is to find w(xyz)\frac{\partial w}{\partial (x y z)}.

To proceed with solving this, let me clarify: are you looking to find the partial derivative of ww with respect to the product xyzxyz, or do you need the gradient vector involving partial derivatives of ww with respect to xx, yy, and zz?

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Math Problem Analysis

Mathematical Concepts

Multivariable Calculus
Partial Derivatives
Chain Rule

Formulas

Partial derivative: ∂w/∂(xyz)
x = y^2 + z^2
y = x + z^2
z = n * y
w = x + y + z

Theorems

Chain Rule for Partial Derivatives

Suitable Grade Level

College level