Final answer:
To find ∂z/∂x and ∂z/∂y for the equation x² + 8y² + 3z² = 4 using implicit differentiation, differentiate both sides of the equation with respect to x and y respectively.
Step-by-step explanation:
To find ∂z/∂x and ∂z/∂y for the equation x² + 8y² + 3z² = 4 using implicit differentiation, we differentiate both sides of the equation with respect to x.
For ∂z/∂x, we treat y and z as functions of x and apply the chain rule, while treating x as an independent variable. Differentiating both sides of the equation with respect to x gives: 2x + 16y * (dy/dx) + 6z * (dz/dx) = 0. Rearranging this equation will give us the value for ∂z/∂x.
Similarly, to find ∂z/∂y, we differentiate both sides of the equation with respect to y treating x and z as functions of y and applying the chain rule. Differentiating both sides of the equation with respect to y gives: 16x * (dx/dy) +16y + 6z * (dz/dy) = 0. Rearranging this equation will give us the value for ∂z/∂y.