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Points: 0 of 1 Find the following derivatives. Express your answer in terms of the independent variables. \[ w_{s} \text { and } w_{t}, \text { where } w=\frac{x-z}{3 y+z}, x=3 s+4 t, y=s t, \text { and } z=3 s-4 t \] ( ype an expression using \( x, y \), and \( z \) as the variadies.) \[ \frac{\partial x}{\partial s}=3 \] (Type an expression using \( s \) and \( t \) as the variables.) \[ \frac{\partial y}{\partial s}=t \] (Type an expression using \( s \) and \( t \) as the variables.) \[ \frac{\partial z}{\partial s}=3 \] (Type an expression using \( s \) and \( t \) as the variables.) \[ \frac{\partial x}{\partial t}=4 \] (Type an expression using \( s \) and \( t \) as the variables.) \[ \frac{\partial y}{\partial t}=\square \] (Type an expression using \( s \) and \( t \) as the variables.)

Ask by Barker Williams. in the United States
Mar 17,2025

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Answer

The derivatives are: - \( \frac{\partial x}{\partial s} = 3 \) - \( \frac{\partial y}{\partial s} = t \) - \( \frac{\partial z}{\partial s} = 3 \) - \( \frac{\partial x}{\partial t} = 4 \) - \( \frac{\partial y}{\partial t} = s \) And the partial derivatives of \( w \) are: - \( w_s = \frac{-(x - z)(3t + 3)}{(3y + z)^2} \) - \( w_t = \frac{(3y + z)(8) - (x - z)(3s - 4)}{(3y + z)^2} \)

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Mind Expander

To find the remaining derivative, we need to calculate \(\frac{\partial y}{\partial t}\) where \(y = s t\). When taking the partial derivative of \(y\) with respect to \(t\), we consider \(s\) as a constant. Therefore, the derivative is: \[ \frac{\partial y}{\partial t} = s \] So the final answer is \(\boxed{s}\).

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