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Let's solve the problem step by step using the chain rule.

We are given:
- \( w = xy + yz + zx \)
- \( x = r \cos(\theta) \)
- \( y = r \sin(\theta) \)
- \( z = r \theta \)

We are tasked to find the partial derivatives \( \frac{\partial w}{\partial r} \) and \( \frac{\partial w}{\partial \theta} \) using the chain rule when \( r = 2 \) and \( \theta = \frac{\pi}{2} \).

### Step 1: Deriving \( w \) in terms of \( r \) and \( \theta \)

Substitute the expressions for \( x \), \( y \), and \( z \) into the equation for \( w \):

\[
w = (r \cos(\theta))(r \sin(\theta)) + (r \sin(\theta))(r \theta) + (r \theta)(r \cos(\theta))
\]

Simplify each term:

\[
w = r^2 \cos(\theta) \sin(\theta) + r^2 \sin(\theta) \theta + r^2 \theta \cos(\theta)
\]

Now, let's calculate the partial derivatives with respect to \( r \) and \( \theta \).

### Step 2: Calculating \( \frac{\partial w}{\partial r} \)

Differentiate \( w \) with respect to \( r \), treating \( \theta \) as constant:

\[
\frac{\partial w}{\partial r} = 2r \cos(\theta) \sin(\theta) + 2r \sin(\theta) \theta + 2r \theta \cos(\theta)
\]

Now substitute \( r = 2 \) and \( \theta = \frac{\pi}{2} \):

- \( \cos\left(\frac{\pi}{2}\right) = 0 \)
- \( \sin\left(\frac{\pi}{2}\right) = 1 \)

So,

\[
\frac{\partial w}{\partial r} = 2(2) \cdot 0 \cdot 1 + 2(2) \cdot 1 \cdot \frac{\pi}{2} + 2(2) \cdot \frac{\pi}{2} \cdot 0
\]

Simplifying this:

\[
\frac{\partial w}{\partial r} = 0 + 4 \cdot \frac{\pi}{2} + 0 = 2\pi
\]

### Step 3: Calculating \( \frac{\partial w}{\partial \theta} \)

Now differentiate \( w \) with respect to \( \theta \), treating \( r \) as constant:

\[
\frac{\partial w}{\partial \theta} = r^2 (\cos(\theta) \cos(\theta) - \sin(\theta) \sin(\theta)) + r^2 (\cos(\theta) \theta - \sin(\theta))
\]

Simplifying further:

We need to apply the product rule. However,

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

Detailed step-by-step solution using the chain rule to find partial derivatives ∂w/∂r and ∂w/∂θ for the function w = xy + yz + zx with x = r cos(θ), y = r sin(θ), and z = rθ. The problem is evaluated at r = 2 and θ = π/2, with calculus concepts explained thoroughly. Suitable for college-level mathematics students.

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