Final answer:
The student is seeking the partial derivatives Zx and Zy for a function z(x, y) implicitly defined by a complex equation. Applying implicit differentiation is the correct approach, yet a specific solution can't be provided without further context or simplification of the equation.
Step-by-step explanation:
The question asks for the partial derivatives Zₓ (denoted by Zx) and Zₒ (denoted by Zy) of the function z(x, y) implicitly defined by the equation 2yz³ + 5x²z² = e³ˣ⁻²ˢ+ᵧ.
To find these partial derivatives, one would apply the method of implicit differentiation. Unfortunately, due to the nature of the function and the provided equation, a specific solution cannot be easily given without additional information. Generally, one takes the partial derivative of both sides of the equation with respect to x (while treating y as a constant) for Zx and with respect to y (while treating x as a constant) for Zy, and then solve for the desired partial derivatives.
The procedure involves differentiating each term and applying the chain rule where necessary. As the provided equation does not lend itself to a simple analytical solution for its partial derivatives, no further detailed steps can be given without additional information or context.