Math Problem Statement

Use the chain rule to find ∂z ∂s and ∂z ∂t . z = x5y3, x = s cos(t), y = s sin(t)

Solution

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Math Problem Analysis

Mathematical Concepts

Multivariable Calculus
Partial Derivatives
Chain Rule

Formulas

∂z/∂s = (∂z/∂x)(∂x/∂s) + (∂z/∂y)(∂y/∂s)
∂z/∂t = (∂z/∂x)(∂x/∂t) + (∂z/∂y)(∂y/∂t)
∂z/∂x = 5x^4 y^3
∂z/∂y = 3x^5 y^2
∂x/∂s = cos(t)
∂y/∂s = sin(t)
∂x/∂t = -s sin(t)
∂y/∂t = s cos(t)

Theorems

Chain Rule in Multivariable Calculus

Suitable Grade Level

Undergraduate Calculus (College Level)