Question

Use chain rule to find the partial derivatives σu/σz and σv/σz of z = e x2y , where x(u,v) = √ uv aand y (u,v) = v/1

Answer 1

To find the partial derivatives σu/σz and σv/σz of z = e^(x^2y), apply the chain rule and simplify the expressions.

Step-by-step explanation:

To find the partial derivatives σu/σz and σv/σz of z = ex^2y, we need to apply the chain rule.
First, we find the partial derivatives of x(u,v) and y(u,v):

x(u,v) = √(uv)
∂x/∂u = (1/2)√(v/u)
∂x/∂v = (1/2)√(u/v)

y(u,v) = v/1
∂y/∂u = 0
∂y/∂v = 1

Next, we use the chain rule to find the partial derivatives of z with respect to u and v:

∂z/∂u = ∂z/∂x * ∂x/∂u + ∂z/∂y * ∂y/∂u

∂z/∂v = ∂z/∂x * ∂x/∂v + ∂z/∂y * ∂y/∂v

We substitute the partial derivatives of x and y into these equations and simplify to find the partial derivatives of z:

σu/σz = e^(x^2y) * ((1/2)√(v/u) * 2xy + 0)

σv/σz = e^(x^2y) * ((1/2)√(u/v) * x^2 + 1)