Question
Part 6 of 11 Find the following derivatives. Express your answer in terms of the independent variables. \( w_{s} \) and \( w_{t} \), where \( w=\frac{x-z}{3 y+z}, x=3 s+4 t, y=s t \), and \( z=3 s-4 t \) \( \frac{\partial w}{\partial y}=-\frac{3(x-z)}{(3 y+z)^{2}} \) \( \frac{\partial w}{\partial z}=-\frac{x+3 y}{(3 y+z)^{2}} \) (Type an expression using \( x, y \), and \( z \) as the variables.) \( \frac{\partial x}{\partial s}=3 \) (Type an expression using \( x, y \), and \( z \) as the variables.) \( \frac{\partial y}{\partial s}=t \) (Type an expression using \( s \) and \( t \) as the variables.) \( \frac{\partial z}{\partial s}=\square \) (Type an expression using \( s \) and \( t \) as the variables.) (Tyiables.)
Ask by Potter Barker. in the United States
Mar 17,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\[
\frac{\partial z}{\partial s} = 3
\]
Solution

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Bonus Knowledge
To find the derivative \( \frac{\partial z}{\partial s} \), we start with the expression for \( z \): \[ z = 3s - 4t \] Taking the partial derivative of \( z \) with respect to \( s \): \[ \frac{\partial z}{\partial s} = \frac{\partial}{\partial s}(3s - 4t) = 3 \] So, the expression is: \[ \frac{\partial z}{\partial s} = 3 \]