Pind the following derivatives. 7 of 8 \( z_{s} \) and \( z_{t} \), where \( z=7 x y-2 x^{2} y, x=2 s+5 t \), and \( y=2 s-5 t \) \( \frac{\partial s}{\partial z}=< \) (Type an expression using s and \( t \) as the variables.) 0.63 of 1 \( \frac{\partial z}{\partial y}=7 x-2 x^{2} \) (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.) \( \frac{\partial y}{\partial t}=-5 \) (Type an expression using \( s \) and \( t \) as the variables.) \( z_{s}=\square \) (Type an expression using \( s \) and \( t \) as the variahloc)

We have \[ z=7xy-2x^2y,\qquad x=2s+5t,\qquad y=2s-5t. \] We want to compute \[ z_s=\frac{\partial z}{\partial s}. \] Using the chain rule we write \[ z_s=\frac{\partial z}{\partial x}\frac{\partial x}{\partial s}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial s}. \] **Step 1. Compute \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\):** Differentiate \(z=7xy-2x^2y\) with respect to \(x\) (treating \(y\) as a constant): \[ \frac{\partial z}{\partial x}=7y-4xy. \] Differentiate \(z\) with respect to \(y\) (treating \(x\) as a constant): \[ \frac{\partial z}{\partial y}=7x-2x^2. \] **Step 2. Use the given derivatives for \(x\) and \(y\):** We have \[ \frac{\partial x}{\partial s}=2\quad \text{and}\quad \frac{\partial y}{\partial s}=2. \] Thus by the chain rule, \[ z_s=(7y-4xy)(2)+(7x-2x^2)(2)=2\left(7y-4xy+7x-2x^2\right). \] **Step 3. Express in terms of \(s\) and \(t\):** Recall that \[ x=2s+5t\quad \text{and}\quad y=2s-5t. \] Substitute these into the expression: 1. Compute individual terms: \[ 7y=7(2s-5t)=14s-35t. \] \[ 4xy=4(2s+5t)(2s-5t)=4\Bigl[(2s)^2-(5t)^2\Bigr]=4(4s^2-25t^2)=16s^2-100t^2. \] \[ 7x=7(2s+5t)=14s+35t. \] \[ 2x^2=2(2s+5t)^2=2(4s^2+20st+25t^2)=8s^2+40st+50t^2. \] 2. Now plug into the bracket: \[ 7y-4xy+7x-2x^2 = (14s-35t)-(16s^2-100t^2)+(14s+35t)-(8s^2+40st+50t^2). \] Combine like terms: - \(s^2\) terms: \[ -16s^2-8s^2=-24s^2. \] - \(st\) term: \[ -40st. \] - \(t^2\) terms: \[ +100t^2-50t^2=+50t^2. \] - \(s\) terms: \[ 14s+14s=28s. \] - \(t\) terms: \[ -35t+35t=0. \] Thus, \[ 7y-4xy+7x-2x^2=28s-24s^2-40st+50t^2. \] Multiply by 2: \[ z_s=2\Bigl(28s-24s^2-40st+50t^2\Bigr)=56s-48s^2-80st+100t^2. \] The final expression for \( z_s \) using \( s \) and \( t \) is: \[ \boxed{-48s^2-80st+56s+100t^2.} \]