Question

Use the chain rule to find the indicated partial derivatives. Given w = xy yz zx, x = r cos(θ), y = r sin(θ), z = r, find ∂w/∂r and ∂w/∂θ when r = 4 and θ = 2.

Answer 1

To find the partial derivatives ∂w/∂r and ∂w/∂θ, we can use the chain rule. Substitute the given values of x, y, and z into the expression for w, and then differentiate with respect to r and θ using the chain rule.

Step-by-step explanation:

To find ∂w/∂r and ∂w/∂θ, we can use the chain rule. First, we need to substitute the given values of x, y, and z into the expression for w. Then, we differentiate with respect to r and θ using the chain rule.

For ∂w/∂r:

w = xy yz zx = (r cos(θ))(r sin(θ))(r)(r cos(θ))

∂w/∂r = yz zx (r sin(θ))(r cos(θ)) + xy zx (r cos(θ))(r sin(θ)) + xy yz (r cos(θ))(r sin(θ))

Substituting r = 4 and θ = 2 into this expression will give you the value of ∂w/∂r.

For ∂w/∂θ:

∂w/∂θ = xy zx (r cos(θ))(-r sin(θ)) + xy yz (-r sin(θ))(r cos(θ)) + xy yz (r cos(θ))(r sin(θ))

Again, substituting r = 4 and θ = 2 will give you the value of ∂w/∂θ.