Use the chain rule to find the indicated partial derivatives.
w = xy + yz + zx,   x = r cos(𝜃),   y = r sin(𝜃),   z = r𝜃

∂w/∂𝜃, when  r = 2, 𝜃 = 𝜋/2


We are given the function \( w = xy + yz + zx \), with \( x = r \cos(\theta), y = r \sin(\theta), z = r \theta \), and are tasked with finding the partial derivative \( \frac{\partial w}{\partial \theta} \) at \( r = 2 \) and \( \theta = \frac{\pi}{2} \).

### Step 1: Apply the chain rule for multivariable functions
We need to differentiate \( w(x, y, z) \) with respect to \( \theta \), considering that \( x, y, z \) are functions of \( \theta \) and \( r \).

\[
\frac{\partial w}{\partial \theta} = \frac{\partial w}{\partial x} \frac{\partial x}{\partial \theta} + \frac{\partial w}{\partial y} \frac{\partial y}{\partial \theta} + \frac{\partial w}{\partial z} \frac{\partial z}{\partial \theta}
\]

### Step 2: Find the partial derivatives of \( w \)

From \( w = xy + yz + zx \), we compute:

- \( \frac{\partial w}{\partial x} = y + z \)
- \( \frac{\partial w}{\partial y} = x + z \)
- \( \frac{\partial w}{\partial z} = x + y \)

### Step 3: Find the partial derivatives of \( x, y, z \) with respect to \( \theta \)

From the parameterizations:

- \( x = r \cos(\theta) \), so \( \frac{\partial x}{\partial \theta} = -r \sin(\theta) \)
- \( y = r \sin(\theta) \), so \( \frac{\partial y}{\partial \theta} = r \cos(\theta) \)
- \( z = r \theta \), so \( \frac{\partial z}{\partial \theta} = r \)

### Step 4: Substitute the expressions

Now we substitute the derivatives into the chain rule expression:

\[
\frac{\partial w}{\partial \theta} = (y + z)(-r \sin(\theta)) + (x + z)(r \cos(\theta)) + (x + y)(r)
\]

### Step 5: Substitute \( x, y, z \) in terms of \( r \) and \( \theta \)

Using \( x = r \cos(\theta), y = r \sin(\theta), z = r \theta \), the expression becomes:

\[
\frac{\partial w}{\partial \theta} = \left( r \sin(\theta) + r \theta \right)(-r \sin(\theta)) + \left( r \cos(\theta) + r \theta \right)(r \cos(\theta)) + \left( r \cos(\theta) + r \sin(\theta) \right)(r)
\]

### Step 6: Evaluate at \( r = 2 \) and \( \theta = \frac{\pi}{2} \)

- \( x = 2 \cos\left(\frac{\pi}{2}\right) = 0 \)
- \( y = 2 \sin\left(\frac{\pi}{2}\right) = 2 \)
- \( z = 2 \cdot \frac{\pi}{2} = \pi \)

Now, substituting into the expression:

\[
\frac{\partial w}{\partial \theta} = \left( 2 + \pi \right)(-2) + \left( 0 + \pi \right)(0) + \left( 0 + 2 \right)(2)
\]

Simplifying:

\[
\frac{\partial w}{\partial \theta} = (2 + \pi)(-2) + 0 + 4
\]
\[
\frac{\partial w}{\partial \theta} = -4 - 2\pi + 4 = -2\pi
\]

### Final Answer:
\[
\frac{\partial w}{\partial \theta} = -2\pi
\]

Would you like more details on any step or further clarification?

Here are 5 related questions:
1. How is the chain rule applied in multivariable calculus?
2. How do you find partial derivatives for implicit functions?
3. Can you explain the significance of the gradient vector in partial derivatives?
4. What happens if \( r \) is not constant in this problem?
5. How does the chain rule generalize to higher dimensions?

**Tip:** Always double-check if all dependent variables are considered when applying the chain rule in multivariable contexts.

Use the chain rule to find the indicated partial derivatives. w = xy + yz + zx, x = r cos(𝜃), y = r sin(𝜃), z = r𝜃 ∂w/∂𝜃, when r = 2, 𝜃 = 𝜋/2

This page explains how to use the chain rule to find partial derivatives of w = xy + yz + zx, with x = r cos(𝜃), y = r sin(𝜃), and z = r𝜃. A step-by-step solution is provided for evaluating ∂w/∂𝜃 when r = 2 and 𝜃 = 𝜋/2. Key concepts include multivariable calculus and the chain rule.

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