Question

Use the chain rule to find the indicated partial derivatives. Let p = u² v² w², where u = xe^y, v = ye^x, and w = exe. Find ∂p/∂x and ∂p/∂y when x = 0 and y = 2.

Answer 1

To find ∂p/∂x and ∂p/∂y, we use the chain rule. After calculating the partial derivatives, we substitute x=0 and y=2 to find the values.

Step-by-step explanation:

To find the partial derivatives of p with respect to x and y, we need to use the chain rule. Let's start by finding ∂p/∂x.

Using the chain rule, we have:

1. Calculate ∂p/∂u = 2²vw²
2. Calculate ∂u/∂x = e^y
3. Calculate ∂v/∂x = ye^x
4. Calculate ∂w/∂x = exe

Multiply these partial derivatives together to get:

∂p/∂x = 2²(xe^y)(ye^x)w²

Now plug in the values x=0 and y=2 to find the value of the partial derivative.