Question

Use the chain rule to find the indicated partial derivatives. Given z = x⁴ * x²y, x = s²t - u, y = stu². Find ∂z/∂s, ∂z/∂t, ∂z/∂u when s = 4, t = 3, u = 2.

Answer 1

To find the partial derivatives, use the chain rule and substitute the given values of s, t, and u into the expressions.

Step-by-step explanation:

To find the partial derivatives, we first need to calculate the derivatives of z with respect to x and y. Using the chain rule, we have:

∂z/∂x = ∂z/∂x * dx/dx + ∂z/∂y * dy/dx

∂z/∂y = ∂z/∂x * dx/dy + ∂z/∂y * dy/dy

∂z/∂s = (∂z/∂x * dx/ds) + (∂z/∂y * dy/ds)

∂z/∂t = (∂z/∂x * dx/dt) + (∂z/∂y * dy/dt)

∂z/∂u = (∂z/∂x * dx/du) + (∂z/∂y * dy/du)

Substituting the given values s = 4, t = 3, u = 2 into the expressions for ∂z/∂s, ∂z/∂t, and ∂z/∂u will give us the final answers.