Final answer:
The partial derivatives ∂z/∂x and ∂z/∂y of the function ez = 6xyz are found using implicit differentiation, yielding ∂z/∂x = (6yz)/ez and ∂z/∂y = (6xz)/ez.
Step-by-step explanation:
To find the partial derivatives ∂z/∂x and ∂z/∂y given the equation ez = 6xyz, we must use implicit differentiation, assuming z is a differentiable function of both x and y. Firstly, for ∂z/∂x, we take the partial derivative of both sides of the equation with respect to x, treating y as a constant:
∂/∂x (ez) = ∂/∂x (6xyz) ⇒ ez∂z/∂x = 6yz + 0 + 0 ⇒ ∂z/∂x = (6yz) / ez. Secondly, for ∂z/∂y, we take the partial derivative of both sides with respect to y, treating x as constant: ∂/∂y (ez) = ∂/∂y (6xyz) ⇒ ez∂z/∂y = 0 + 6xz + 0 ⇒ ∂z/∂y = (6xz) / ez.
Therefore, the partial derivatives of z with respect to x and y are ∂z/∂x = (6yz) / ez and ∂z/∂y = (6xz) / ez, respectively.