\( 1 \leftarrow \quad \) Find the following derivatives. \( z_{s} \) and \( z_{4} \), where \( z=7 x y-2 x^{2} y, x=2 s+5 t \), and \( y=2 s-5 t \) \( \frac{\partial z}{\partial x}=\square \) (Type an expression using \( x \) and \( y \) as the variables.)

To find the partial derivative \( \frac{\partial z}{\partial x} \), we start with the expression for \( z \): \[ z = 7xy - 2x^2y \] Taking the partial derivative with respect to \( x \): \[ \frac{\partial z}{\partial x} = 7y - 4xy \] Now, substituting \( x \) and \( y \): \[ x = 2s + 5t \quad \text{and} \quad y = 2s - 5t \] Let's plug these into the derivative: \[ \frac{\partial z}{\partial x} = 7(2s - 5t) - 4(2s + 5t)(2s - 5t) \] Now you have an expression involving \( s \) and \( t \) corresponding to the variables \( x \) and \( y \). Enjoy exploring the relationships!