Texts: Use the Chain Rule to find the indicated partial derivatives.
z = x + x²y, x = s + 2t - u, y = stu²;
∂z/∂s = ?
∂z/∂t = ?
∂z/∂u = ?
Given: s = 5, t = 1, u = 3
Solution:
To find ∂z/∂s, we need to find ∂z/∂x, ∂x/∂s, and ∂y/∂s.
∂z/∂x = 1 + 2xy
∂x/∂s = 1
∂y/∂s = tu²
∂z/∂s = (∂z/∂x) * (∂x/∂s) + (∂z/∂y) * (∂y/∂s)
= (1 + 2xy) * 1 + 0 * tu²
= 1 + 2xy
Substituting the given values:
∂z/∂s = 1 + 2(5)(1)(3)
= 1 + 30
= 31
Therefore, ∂z/∂s = 31.
To find ∂z/∂t, we need to find ∂z/∂x, ∂x/∂t, and ∂y/∂t.
∂z/∂x = 1 + 2xy
∂x/∂t = 2
∂y/∂t = su²
∂z/∂t = (∂z/∂x) * (∂x/∂t) + (∂z/∂y) * (∂y/∂t)
= (1 + 2xy) * 2 + 0 * su²
= 2 + 4xy
Substituting the given values:
∂z/∂t = 2 + 4(5)(1)(3)
= 2 + 60
= 62
Therefore, ∂z/∂t = 62.
To find ∂z/∂u, we need to find ∂z/∂x, ∂x/∂u, and ∂y/∂u.
∂z/∂x = 1 + 2xy
∂x/∂u = -1
∂y/∂u = -st
∂z/∂u = (∂z/∂x) * (∂x/∂u) + (∂z/∂y) * (∂y/∂u)
= (1 + 2xy) * (-1) + 0 * (-st)
= -1 - 2xy
Substituting the given values:
∂z/∂u = -1 - 2(5)(1)(3)
= -1 - 30
= -31
Therefore, ∂z/∂u = -31.
Hence, the partial derivatives are:
∂z/∂s = 31
∂z/∂t = 62
∂z/∂u = -31.