Final answer:
For the equation x² + 2y² + 7z² = 1, ∂z/∂x is -x/(7z) and ∂z/∂y is -2y/(7z) after applying partial differentiation with respect to x and y, respectively.
Step-by-step explanation:
To find ∂z/∂x and ∂z/∂y for the given equation x² + 2y² + 7z² = 1, we need to use partial differentiation. Partial differentiation involves differentiating with respect to one variable while keeping the other variables constant.
For ∂z/∂x, differentiate the equation with respect to x, treating y and z as constants:
2x + 0 + 0 = 0 (since the derivative of y and z terms with respect to x are zero)
∂z/∂x = -2x/(14z) (after rearranging and solving for ∂z/∂x)
For ∂z/∂y, differentiate the equation with respect to y, treating x and z as constants:
0 + 4y + 0 = 0 (since the derivative of x and z terms with respect to y are zero)
∂z/∂y = -4y/(14z) (after rearranging and solving for ∂z/∂y)
To find ∂z/∂x and ∂z/∂y for the equation x² + 2y² + 7z² = 1, we need to differentiate the equation with respect to x and y.
When we differentiate x² + 2y² + 7z² = 1 with respect to x, we get 2x + 0 + 0 = 0. Therefore, ∂z/∂x = 0.
When we differentiate x² + 2y² + 7z² = 1 with respect to y, we get 0 + 4y + 0 = 0. Therefore, ∂z/∂y = 0.