Question

Use the chain rule to find the indicated partial derivatives. Given w = xy yz zx, x = r cosθ, y = r sinθ, z = r. Find ∂w/∂r when r = 2 and θ = 2.

Answer 1

To find the partial derivative ∂w/∂r for w = xy yz zx with the given parametric equations, we express w in terms of r and θ, apply the chain rule, and then substitute the values of r = 2 and θ = 2 to calculate the derivative at the specified point.

Step-by-step explanation:

To calculate the partial derivative of w with respect to r when w = xy yz zx, and x = r cosθ, y = r sinθ, z = r, we first need to express w in terms of r and θ:

• w = (r cosθ) (r sinθ) (r)

Now, simplify the expression:

• w = r³ cosθ sinθ

To find ∂w/∂r, we apply the chain rule:

• ∂w/∂r = 3r² cosθ sinθ

Plugging in the given values of r = 2 and θ = 2:

• ∂w/∂r |r=2, θ2 = 3(2)² cos(2) sin(2)

Now, calculate the numerical value to get the final answer.