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 = 1, t = 2, u = 4.

Answer 1

To find the partial derivatives of z with respect to s, t, and u, we need to apply the chain rule. The partial derivatives are ∂z/∂s = 32, ∂z/∂t = -4, and ∂z/∂u = -32 when s = 1, t = 2, and u = 4.

Step-by-step explanation:

To find the partial derivatives of z with respect to s, t, and u, we need to apply the chain rule. Let's start by finding ∂z/∂s:

∂z/∂s = (∂z/∂x) * (∂x/∂s)

∂z/∂s = (4x³ * x²y) * (2st)

∂z/∂s = 8x⁵ * sty

Now, let's find ∂z/∂t:

∂z/∂t = (∂z/∂x) * (∂x/∂t)

∂z/∂t = (4x³ * x²y) * (s² - u)

∂z/∂t = 4x³ * x²y * (s² - u)

Finally, let's find ∂z/∂u:

∂z/∂u = (∂z/∂x) * (∂x/∂u)

∂z/∂u = (4x³ * x²y) * (-1)

∂z/∂u = -4x³ * x²y

For s = 1, t = 2, and u = 4, we substitute these values into the partial derivatives to evaluate them.

∂z/∂s = 8(1⁵) * (1 * 2 * 2) = 32

∂z/∂t = 4(1³) * (1² * 2 * 2 - 4) = -4

∂z/∂u = -4(1³) * (1² * 2 * 2) = -321