Question

Use the chain rule to find ∂z/∂s and ∂z/∂t. Given z = (x - y)⁹, x = s²t, and y = st². Find ∂z/∂s and ∂z/∂t.

Answer 1

To find ∂z/∂s and ∂z/∂t, we use the chain rule. ∂z/∂s = 18t²(s²t - st²)⁸(2s³ - t³) and ∂z/∂t = 18s³(s²t - st²)⁸(s² - 2t²).

Step-by-step explanation:

To find ∂z/∂s and ∂z/∂t, we will use the chain rule. Let's start by finding ∂z/∂s:

1. Using the given information, we have x = s²t and y = st².
2. Substituting these values into z = (x - y)⁹, we get z = (s²t - st²)⁹.
3. Taking the partial derivative of z with respect to s, we apply the chain rule:

• ∂z/∂s = ∂z/∂x * ∂x/∂s + ∂z/∂y * ∂y/∂s.
• ∂z/∂x = 9(s²t - st²)⁸ * 2st = 18s²t(s²t - st²)⁸.
• ∂x/∂s = 2st.
• ∂z/∂y = 9(s²t - st²)⁸ * (-2t²) = -18t²(s²t - st²)⁸.
• ∂y/∂s = t².
• Substituting these values into ∂z/∂s = ∂z/∂x * ∂x/∂s + ∂z/∂y * ∂y/∂s, we get ∂z/∂s = 18s²t(s²t - st²)⁸ * 2st - 18t²(s²t - st²)⁸ * t².
• Simplifying further, ∂z/∂s = 36s³t²(s²t - st²)⁸ - 18t³(s²t - st²)⁸ = 18t²(s²t - st²)⁸(2s³ - t³).

Now let's find ∂z/∂t:

1. Using the given information, we have x = s²t and y = st².
2. Substituting these values into z = (x - y)⁹, we get z = (s²t - st²)⁹.
3. Taking the partial derivative of z with respect to t, we apply the chain rule:

• ∂z/∂t = ∂z/∂x * ∂x/∂t + ∂z/∂y * ∂y/∂t.
• ∂z/∂x = 9(s²t - st²)⁸ * 2s = 18s(s²t - st²)⁸.
• ∂x/∂t = s².
• ∂z/∂y = 9(s²t - st²)⁸ * (-2st) = -18s²t(s²t - st²)⁸.
• ∂y/∂t = 2st.
• Substituting these values into ∂z/∂t = ∂z/∂x * ∂x/∂t + ∂z/∂y * ∂y/∂t, we get ∂z/∂t = 18s(s²t - st²)⁸ * s² - 18s²t(s²t - st²)⁸ * 2st.
• Simplifying further, ∂z/∂t = 18s³(s²t - st²)⁸(s² - 2t²).