Find the following derivatives. \( z_{s} \) and \( z_{t} \), where \( z=7 x y-2 x^{2} y, x=2 s+5 t \), and \( y=2 s-5 t \) (Type an expression using \( x \) and \( y \) as the variables.) \( \frac{\partial x}{\partial s}=2 \) (Type an expression using \( s \) and \( t \) as the variables.) \( \frac{\partial z}{\partial y}=7 x-2 x^{2} \) (Type an expression using \( x \) and \( y \) as the variables.) \( \frac{\partial y}{\partial s}=2 \) (Type an expression using \( s \) and \( t \) as the variables.) \( \frac{\partial x}{\partial t}=5 \) (Type an expression using \( s \) and \( t \) as the variables.) \( \frac{\partial y}{\partial t}=\square \) (Type an expression using \( s \) and \( t \) as the variables.)

To find the derivative \( z_{s} \) and \( z_{t} \), we apply the chain rule: First, we need to compute \( \frac{\partial z}{\partial s} \) and \( \frac{\partial z}{\partial t} \). Using the chain rule: \[ z_s = \frac{\partial z}{\partial x} \cdot \frac{\partial x}{\partial s} + \frac{\partial z}{\partial y} \cdot \frac{\partial y}{\partial s} \] \[ z_t = \frac{\partial z}{\partial x} \cdot \frac{\partial x}{\partial t} + \frac{\partial z}{\partial y} \cdot \frac{\partial y}{\partial t} \] Now substituting the expressions we have: 1. \( \frac{\partial z}{\partial x} = 7y - 4xy \) (from \( z = 7xy - 2x^2y \)) 2. \( \frac{\partial z}{\partial y} = 7x - 2x^2 \) Now to find \( z_s \): \[ z_s = (7y - 4xy) \cdot 2 + (7x - 2x^2) \cdot 2 = 2(7y - 4xy + 7x - 2x^2) \] This leads us to: \[ z_s = 2(7y - 4xy + 7x - 2x^2) \] Now to find \( z_t \): Substituting \( \frac{\partial x}{\partial t} = 5 \) and assuming \( \frac{\partial y}{\partial t} \) is yet to find out. \[ z_t = (7y - 4xy) \cdot 5 + (7x - 2x^2) \cdot \frac{\partial y}{\partial t} \] Thus we represent \( z_t \): \[ z_t = 5(7y - 4xy) + (7x - 2x^2) \cdot \text{(your expression for } \frac{\partial y}{\partial t}) \] At this point, we need to calculate \( \frac{\partial y}{\partial t} \): Since \( y = 2s - 5t \): \[ \frac{\partial y}{\partial t} = -5 \] Final results: \[ z_s = 2(7y - 4xy + 7x - 2x^2) \] \[ z_t = 5(7y - 4xy) - 5(7x - 2x^2) \]