Question

Use the Chain Rule to find the indicated partial derivatives for the following functions:

z = x^4 + x^2y, x = s + 2t - u, y = stu^2

1. ∂z/∂s
2. ∂z/∂t
3. ∂z/∂u

Evaluate when s = 3, t = 2, u = 1

Answer 1

To find the partial derivatives of the function z = x^4 + x^2y with respect to s, t, and u, we can use the chain rule. The partial derivatives are ∂z/∂s = (4x^3 + 2xy) + x^2 * tu^21, ∂z/∂t = (4x^3 + 2xy) * 2 + x^2 * su^21, and ∂z/∂u = (4x^3 + 2xy) * (-1) + x^2 * st(21)u^20. When s = 3, t = 2, and u = 1, substitute these values into the derivatives to evaluate them.

Step-by-step explanation:

To find the indicated partial derivatives of the function z = x^4 + x^2y with respect to s, t, and u, we can use the chain rule. First, we need to find the partial derivatives of x and y with respect to s, t, and u.

1. ∂x/∂s = 1
2. ∂x/∂t = 2
3. ∂x/∂u = -1
4. ∂y/∂s = tu^21
5. ∂y/∂t = su^21
6. ∂y/∂u = st(21)u^20

Next, we can use these partial derivatives to find the desired partial derivatives of z.

1. ∂z/∂s = ∂z/∂x * ∂x/∂s + ∂z/∂y * ∂y/∂s = (4x^3 + 2xy) * 1 + x^2 * tu^21
2. ∂z/∂t = ∂z/∂x * ∂x/∂t + ∂z/∂y * ∂y/∂t = (4x^3 + 2xy) * 2 + x^2 * su^21
3. ∂z/∂u = ∂z/∂x * ∂x/∂u + ∂z/∂y * ∂y/∂u = (4x^3 + 2xy) * (-1) + x^2 * st(21)u^20

To evaluate these partial derivatives when s = 3, t = 2, and u = 1, substitute these values into the derivatives obtained in step 3.