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 = 3, t = 4, u = 1.

Answer 1

∂z/∂s = 8t(3²)(4) - 1 = 192t - 1

∂z/∂t = 8s(4)²(3) - u = 96s - u

∂z/∂u = -8s(4)²(3) + 1 = -96s + 1

Step-by-step explanation:

To find ∂z/∂s, we use the chain rule. First, we find ∂z/∂x, which is 4x³ x²y. Then, we find ∂x/∂s, which is 2st. Substituting these into the chain rule formula, we get ∂z/∂s = (4x³ x²y)(2st) = 8t(x⁴)(3²)(4) - y = 192t - y. Substituting the given values for x and y, we get ∂z/∂s = 192t - 1.

For ∂z/∂t, we again start by finding ∂z/∂x, which is 4x³ x²y. Then, we find ∂x/∂t, which is 2su. Substituting these into the chain rule formula, we get ∂z/∂t = (4x³ x²y)(2su) = 8s(x⁴)(4)²(3) - u = 96s - u.

Finally, for ∂z/∂u, we find ∂z/∂x as before and then calculate ∂x/∂u, which is -1. Substituting these into the chain rule formula gives us ∂z/∂u = (4x³ * x²y)(-1) = -8s(x⁴)(4)²(3) + 1 = -96s + 1.