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 = 5, u = 4.

Answer 1

To find the indicated partial derivatives ∂z/∂s, ∂z/∂t, and ∂z/∂u, we can use the chain rule. First, find ∂z/∂x and ∂z/∂y using the power rule of differentiation. Then, substitute x = s²t - u and y = stu², and evaluate the partial derivatives with the given values of s, t, and u.

Step-by-step explanation:

To find the partial derivatives ∂z/∂s, ∂z/∂t, and ∂z/∂u, we can use the chain rule. Let's start with ∂z/∂s. Using the chain rule, we have ∂z/∂s = (∂z/∂x)(∂x/∂s) + (∂z/∂y)(∂y/∂s).

First, we find ∂z/∂x and ∂z/∂y. Since z = x⁴ * x²y, we can differentiate each term using the power rule of differentiation. ∂z/∂x = 4x³ * x²y + x⁴(2y) = 4x⁵y + 2x⁴y.

Similarly, ∂z/∂y = x⁴ * 2xy = 2x⁴y². Now, we substitute x = s²t - u and y = stu² into the partial derivatives:

∂z/∂s = (4x⁵y + 2x⁴y)(∂(s²t - u)/∂s) + (2x⁴y²)(∂(stu²)/∂s).

∂(s²t - u)/∂s = 2st and ∂(stu²)/∂s = tu². Substitute x = s²t - u, y = stu², and the values s = 3, t = 5, u = 4 into the equation to find the value of ∂z/∂s.

Similarly, we can find ∂z/∂t and ∂z/∂u using the same process. Finally, we substitute s = 3, t = 5, u = 4 into the equations to find the values of ∂z/∂t and ∂z/∂u.