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

Ask by Carter Adkins. in the United States
Mar 17,2025

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\( z_s = 14y - 8xy + 14x - 4x^2 \) \( z_t = 35y - 20xy - 35x + 10x^2 \) \( \frac{\partial x}{\partial t} = 5 \)

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To find \( \frac{\partial x}{\partial t} \), we start with the expression for \( x \): \[ x = 2s + 5t \] Taking the partial derivative with respect to \( t \) gives us: \[ \frac{\partial x}{\partial t} = 5 \] Now, let's summarize the derivatives found: 1. \( \frac{\partial z}{\partial x} = 7y - 4xy \) 2. \( \frac{\partial z}{\partial y} = 7x - 2x^2 \) 3. \( \frac{\partial x}{\partial s} = 2 \) 4. \( \frac{\partial y}{\partial s} = 2 \) 5. \( \frac{\partial x}{\partial t} = 5 \)

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