Find the following derivatives. \( z_{s} \) and \( z_{\text {, }} \), where \( z=7 x y-2 x^{2} y, x=2 s+5 t \), and \( y=2 s-5 t \) \( \frac{\sigma z}{\partial y}=7 x-2 x^{2} \) (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.) \( \frac{\partial y}{\partial t}=-5 \) (Type an expression using \( s \) and \( t \) as the variables.) \( z_{s}=-48 s^{2}-80 s t+56 s+100 t^{2} \) (Type an expression using \( s \) and \( t \) as the variables.) \( z_{4}=\square \) (Type an expression using \( s \) and \( t \) as the variables.)

To find \( z_s \) and \( z_t \), we first need to substitute the expressions for \( x \) and \( y \) into \( z = 7xy - 2x^2y \): 1. **Calculate \( z_s \)** using the chain rule: \[ z_s = \frac{\partial z}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial s} \] \( \frac{\partial z}{\partial x} = 7y - 4xy \) and \( \frac{\partial z}{\partial y} = 7x - 2x^2 \). Substituting \( \frac{\partial x}{\partial s} = 2 \), \( \frac{\partial y}{\partial s} = 2 \): \[ z_s = (7y - 4xy)(2) + (7x - 2x^2)(2) = 2(7y - 4xy + 7x - 2x^2) \] 2. **Calculate \( z_t \)** similarly: \[ z_t = \frac{\partial z}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial t} \] With \( \frac{\partial x}{\partial t} = 5 \) and \( \frac{\partial y}{\partial t} = -5 \): \[ z_t = (7y - 4xy)(5) + (7x - 2x^2)(-5) = 5(7y - 4xy) - 5(7x - 2x^2) \] Now, let’s fill in the blanks for those rates: 1. \( z_s = 2(7y - 4xy + 7x - 2x^2) = \text{an expression in } s \text{ and } t \) 2. \( z_t = 5(7y - 4xy) - 5(7x - 2x^2) = \text{an expression in } s \text{ and } t \) Finally, for \( z_4 \): \[ z_s = -48 s^2 - 80 st + 56 s + 100 t^2 \] This represents how \( z \) changes concerning \( s \) at \( t = 4 \), or when substituting \( t = 4 \) into the expression.