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,