Use the Chain Rule to find the indicated partial derivatives. w = xy + yz + zx, x = r cos(θ), y = r sin(θ), z = rθ; âw âr , âw âθ when r = 4, θ = Ï2

δw/δr = 4π

δw/δθ = -8π

Step-by-step explanation:

Given the following functions

w = xy + yz + zx, x = r cosθ y = r sinθ, z = rθ where r = 4 and θ = π/2

We are to find δw/δr and δw/δθ

δw/δr = δw/δx•δx/δr + δw/δy•δy/δr + δw/δz•δz/δr

δw/δx = y+z

δx/δr = cosθ

δw/δy = x+z

δy/δr = sinθ

δw/δz = y+x

δz/δr = θ

Substitute the given values into the formula

δw/δr = (y+z)cosθ+(x+z)sinθ+(y+x)θ

Substitute the value of x, y and z in terms of theta into the resulting function

δw/δr = (y+z)cosθ+(x+z)sinθ+(y+x)θ

δw/δr = (rsinθ+rθ)cosθ+(rcosθ+rθ)sinθ+(rsinθ+rcosθ)θ

Substitute r = 4 and θ = π/2

δw/δr = (4sinπ/2+4π/2)cosπ/2+(4cosπ/2+4π/2)sinπ/2+(4sinπ/2+4cosπ/2)π/2

Note that cos π/2 = 0 and sinπ/2 = 1

δw/δr = (4+2π)(0)+(0+2π)(1)+(4(1)+4(0))π/2

δw/δr = 0+2π+4π/2

δw/δr = 2π+2π

δw/δr = 4π

For δw/δθ

δw/δθ = δw/δx•δx/δθ + δw/δy•δy/δθ + δw/δz•δz/δθ

δw/δx = y+z

δx/δθ = -rsinθ

δw/δy = x+z

δy/δθ =rcosθ

δw/δz = y+x

δz/δθ = r

Substitute the given values into the formula

δw/δθ = (y+z)-rsinθ+(x+z)rcosθ+(y+x)r

Substitute the value of x, y and z in terms of theta into the resulting function

δw/δθ = (y+z)-rsinθ+(x+z)rcosθ+(y+x)r

δw/δθ = (rsinθ+rθ)-rsinθ+(rcosθ+rθ)rcosθ+(rsinθ+rcosθ)r

Substitute r = 4 and θ = π/2

δw/δθ = (4sinπ/2+4π/2)-4sinπ/2+(4cosπ/2+4π/2)4cosπ/2+(4sinπ/2+4cosπ/2)(4)

Note that cos π/2 = 0 and sinπ/2 = 1

δw/δθ = (4+2π)(-4)+(0+2π)(0)+(4(1)+4(0))(4)

δw/δθ = -16-8π+0+4(4)

δw/δθ = -16+16-8π

δw/δθ = -8π

5+2x=a

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