Question

If z=xey, x=u3-v3, y=u3+v3, find ∂z/∂u and ∂z/∂v. The variables are restricted to domains on which the functions are defined.

Answer 1

`\((\partial z)/(\partial u) = 3u^2(xe^y + e^y)\)` and
`\((\partial z)/(\partial v) = -3v^2(xe^y - e^y)\)`, considering the given expressions for
`\(z\), \(x\)`, and
`\(y\)`.

Step-by-step explanation:

To find
`\((\partial z)/(\partial u)\)` and
`\((\partial z)/(\partial v)\)`, we'll use the chain rule and compute the partial derivatives step by step:

Given:

`\[ z = x \cdot e^y \]`

`\[ x = u^3 - v^3 \]`

`\[ y = u^3 + v^3 \]`

1. Compute
`\((\partial z)/(\partial u)\)`:

`\[ (\partial z)/(\partial u) = (\partial z)/(\partial x) \cdot (\partial x)/(\partial u) + (\partial z)/(\partial y) \cdot (\partial y)/(\partial u) \]`

`\[ (\partial z)/(\partial x) = e^y \]`

`\[ (\partial x)/(\partial u) = 3u^2 \]`

`\[ (\partial z)/(\partial y) = x \cdot e^y \]`

`\[ (\partial y)/(\partial u) = 3u^2 \]`

`\[ (\partial z)/(\partial u) = e^y \cdot 3u^2 + x \cdot e^y \cdot 3u^2 \]`

2. **Compute
`\((\partial z)/(\partial v)\)`:**

`\[ (\partial z)/(\partial v) = (\partial z)/(\partial x) \cdot (\partial x)/(\partial v) + (\partial z)/(\partial y) \cdot (\partial y)/(\partial v) \]`

`\[ (\partial x)/(\partial v) = -3v^2 \]`

`\[ (\partial y)/(\partial v) = 3v^2 \]`

`\[ (\partial z)/(\partial v) = e^y \cdot (-3v^2) + x \cdot e^y \cdot 3v^2 \]`

These expressions provide the partial derivatives of
`\(z\)` with respect to
`\(u\)` and
`\(v\)`, respectively. The specific values will depend on the values of
`\(u\)` and
`\(v\)` in the given domains where the functions are defined.