Find the following derivatives. \( z_{s} \) and \( z_{\text {}} \), 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 \( z_s \) and \( z_t \) for the given expressions, we can utilize the chain rule. Starting with \( z = 7xy - 2x^2y \), we first compute \( z_s \): \[ z_s = \frac{\partial z}{\partial x} \cdot \frac{\partial x}{\partial s} + \frac{\partial z}{\partial y} \cdot \frac{\partial y}{\partial s} \] Using the information provided: 1. \( \frac{\partial z}{\partial y} = 7x - 2x^2 \) 2. \( \frac{\partial x}{\partial s} = 2 \) 3. \( \frac{\partial y}{\partial s} = 2 \) We need \(\frac{\partial z}{\partial x}\): \[ \frac{\partial z}{\partial x} = 7y - 4xy \] Now we can compute: \[ z_s = (7y - 4xy) \cdot 2 + (7x - 2x^2) \cdot 2 \] \[ = 2(7y - 4xy + 7x - 2x^2) \] Next, for \( z_t \): We follow the same approach with the chain rule: \[ z_t = \frac{\partial z}{\partial x} \cdot \frac{\partial x}{\partial t} + \frac{\partial z}{\partial y} \cdot \frac{\partial y}{\partial t} \] From the given data: 1. \( \frac{\partial x}{\partial t} = 5 \) 2. We need to find \( \frac{\partial y}{\partial t} \): Solving for \(\frac{\partial y}{\partial t}\), since \( y = 2s - 5t \): \[ \frac{\partial y}{\partial t} = -5 \] Thus, we can write: \[ z_t = (7y - 4xy) \cdot 5 + (7x - 2x^2) \cdot (-5) \] \[ = 5(7y - 4xy - 7x + 2x^2) \] So, the final outputs are: **Result for \( z_s \)**: \( z_s = 2(7y - 4xy + 7x - 2x^2) \) **Result for \( z_t \)**: \( z_t = 5(7y - 4xy - 7x + 2x^2) \)