Question

Consider the equation 7 sine (x plus y )plus 9 sine (x plus z )plus 2 sine (y plus z )equals0. Find the values of StartFraction partial derivative z Over partial derivative x EndFraction and StartFraction partial derivative z Over partial derivative y EndFraction at the point (pi comma 3 pi comma negative 3 pi ).

Answer 1

(∂z/∂x) = -16/11

(∂z/∂y) = -9/11

Explanation:

7 sin (x + y) + 9 sin (x + z) + 2 sin (y + z) = 0

To obtain ∂z/∂x, at the point (π, 3π, -3π), we will implicitly differentiate the given equation, taking y to be a constant.

7 cos (x + y) + 9 cos (x + z) + (9 cos (x + z)) (∂z/∂x) + 2 cos (y + z) (∂z/∂x) = 0

[9 cos (x + z) + 2 cos (y + z)] (∂z/∂x) = - [7 cos (x + y) + 9 cos (x + z)]

(∂z/∂x) = - [7 cos (x + y) + 9 cos (x + z)]/[9 cos (x + z) + 2 cos (y + z)]

At the point (π, 3π, -3π)

(∂z/∂x) = - [7 cos (π + 3π) + 9 cos (π - 3π)]/[9 cos (π - 3π) + 2 cos (3π - 3π)]

(∂z/∂x) = - [7 cos 4π + 9 cos (-2π)]/(9 cos (-2π) + 2cos 0)

(∂z/∂x) = - (7(1) + 9(1))/(9(1) + 2(1))

(∂z/∂x) = -(7 + 9)/(9 + 2) = - 16/11

b) 7 sin (x + y) + 9 sin (x + z) + 2 sin (y + z) = 0

To obtain ∂z/∂y, at the point (π, 3π, -3π), we will implicitly differentiate the given equation, taking x to be a constant.

7 cos (x + y) + (9 cos (x + z)) (∂z/∂y) + 2 cos (y + z) + 2 cos (y + z) (∂z/∂y) = 0

[9 cos (x + z) + 2 cos (y + z)] (∂z/∂x) = - [7 cos (x + y) + 2 cos (y + z)]

(∂z/∂y) = - [7 cos (x + y) + 2 cos (y + z)]/[9 cos (x + z) + 2 cos (y + z)]

At the point (π, 3π, -3π)

(∂z/∂y) = - [7 cos (π + 3π) + 2 cos (3π - 3π)]/[9 cos (π - 3π) + 2 cos (3π - 3π)]

(∂z/∂y) = - [7 cos 4π + 2 cos (0)]/(9 cos (-2π) + 2cos 0)

(∂z/∂y) = - (7(1) + 2(1))/(9(1) + 2(1))

(∂z/∂y) = -(7 + 2)/(9 + 2) = - 9/11