Use the Chain Rule to find the indicated partial derivatives:
∂z/∂u = (∂z/∂x)(∂x/∂u) + (∂z/∂y)(∂y/∂u)
∂z/∂v = (∂z/∂x)(∂x/∂v) + (∂z/∂y)(∂y/∂v)
∂z/∂w = (∂z/∂x)(∂x/∂w) + (∂z/∂y)(∂y/∂w)
Given:
x = uv^2 + w^4
y = u + v*e^w
z = 2x^2 + y^4
To find ∂z/∂u:
∂z/∂u = (∂z/∂x)(∂x/∂u) + (∂z/∂y)(∂y/∂u)
To find ∂z/∂v:
∂z/∂v = (∂z/∂x)(∂x/∂v) + (∂z/∂y)(∂y/∂v)
To find ∂z/∂w:
∂z/∂w = (∂z/∂x)(∂x/∂w) + (∂z/∂y)(∂y/∂w)
Given values:
u = 1
v = 15
w = 0
To find ∂z/∂u, ∂z/∂v, and ∂z/∂w, we need to calculate the partial derivatives of x and y with respect to u, v, and w, and substitute the given values.