Question

Use the chain rule to find the indicated partial derivatives. Given p = u² * v² * w², u = x * eʸ, v = y * eˣ, w = eˣy. Find ∂p/∂x and ∂p/∂y when x = 0 and y = 4.

Answer 1

To find the partial derivatives ∂p/∂x and ∂p/∂y of p = u² * v² * w², where u = x * eʸ, v = y * eˣ, and w = eˣy, we use the chain rule. Substituting the given values, we calculate the partial derivatives and evaluate them at x = 0 and y = 4.

Step-by-step explanation:

To find the partial derivatives ∂p/∂x and ∂p/∂y, we need to use the chain rule. Starting with p = u² * v² * w², where u = x * eʸ, v = y * eˣ, and w = eˣy, we can calculate:

1. ∂p/∂x = ∂p/∂u * ∂u/∂x + ∂p/∂v * ∂v/∂x + ∂p/∂w * ∂w/∂x
2. ∂p/∂y = ∂p/∂u * ∂u/∂y + ∂p/∂v * ∂v/∂y + ∂p/∂w * ∂w/∂y

Substituting the given values, x = 0 and y = 4, we get:

1. ∂p/∂x = 2 * (u * v² * w²)
2. ∂p/∂y = 2 * (u² * v * w²)

Plugging in the values of u, v, and w, we can evaluate the partial derivatives at x = 0 and y = 4.