Find the following derivatives. \( z_{s} \) and \( z_{\mathrm{t}} \), where \( z=9 x y-3 x^{2} y, x=3 s+5 t \), and \( y=3 s-5 t \) \( z_{s}=-243 s^{2}+162 s-270 s t+225 t^{2} \) (Type an expression using \( s \) and \( t \) as the variables.) \( z_{t}=\square \) (Type an expression using \( s \) and \( t \) as the variables.)

To find the derivatives \( z_s \) and \( z_t \) for the function \( z = 9xy - 3x^2y \) with the substitutions \( x = 3s + 5t \) and \( y = 3s - 5t \), we will use the chain rule. ### Step 1: Calculate \( z_s \) We already have \( z_s = -243s^2 + 162s - 270st + 225t^2 \). ### Step 2: Calculate \( z_t \) To find \( z_t \), we will first need to compute the partial derivatives of \( z \) with respect to \( x \) and \( y \), and then apply the chain rule. 1. **Calculate \( \frac{\partial z}{\partial x} \)**: \[ \frac{\partial z}{\partial x} = 9y - 6xy \] 2. **Calculate \( \frac{\partial z}{\partial y} \)**: \[ \frac{\partial z}{\partial y} = 9x - 3x^2 \] 3. **Calculate \( \frac{\partial x}{\partial t} \)** and \( \frac{\partial y}{\partial t} \)**: \[ \frac{\partial x}{\partial t} = 5, \quad \frac{\partial y}{\partial t} = -5 \] 4. **Apply the chain rule**: \[ z_t = \frac{\partial z}{\partial x} \cdot \frac{\partial x}{\partial t} + \frac{\partial z}{\partial y} \cdot \frac{\partial y}{\partial t} \] Now, substituting the expressions we found: \[ z_t = (9y - 6xy) \cdot 5 + (9x - 3x^2) \cdot (-5) \] ### Step 3: Substitute \( x \) and \( y \) Now we substitute \( x = 3s + 5t \) and \( y = 3s - 5t \) into the expression for \( z_t \): 1. Substitute \( y \): \[ 9y = 9(3s - 5t) = 27s - 45t \] \[ 6xy = 6(3s + 5t)(3s - 5t) = 6(9s^2 - 25t^2) = 54s^2 - 150t^2 \] 2. Substitute \( x \): \[ 9x = 9(3s + 5t) = 27s + 45t \] \[ 3x^2 = 3(3s + 5t)^2 = 3(9s^2 + 30st + 25t^2) = 27s^2 + 90st + 75t^2 \] ### Step 4: Combine the terms Now we can combine everything into the expression for \( z_t \): \[ z_t = (27s - 45t - (54s^2 - 150t^2)) \cdot 5 + (27s + 45t - (27s^2 + 90st + 75t^2)) \cdot (-5) \] Calculating this step by step: 1. First term: \[ (27s - 45t - 54s^2 + 150t^2) \cdot 5 = 135s - 225t - 270s^2 + 750t^2 \] 2. Second term: \[ -(27s + 45t - 27s^2 - 90st - 75t^2) \cdot 5 = -135s - 225t + 135s^2 + 450st + 375t^2 \] ### Step 5: Combine both terms Now, combine the two results: \[ z_t = (135s - 225t - 270s^2 + 750t^2) + (-135s - 225t + 135s^2 + 450st + 375t^2) \] Combining like terms: - For \( s^2 \): \( -270s^2 + 135s^2 = -135s^2 \) - For \( t^2 \): \( 750t^2 + 375t^2 = 1125t^2 \) - For \( st \): \( 450st \) - For \( s \): \( 0 \) - For \( t \): \( -450t \) Thus, we have: \[ z_t = -135s^2 + 450st + 1125t^2 - 450t \] ### Final Answer So, the expression for \( z_t \) is: \[ z_t = -135s^2 + 450st + 1125t^2 - 450t \]