Part 10 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 \( z=3 s-4 t \) os 0 of 1 (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.) \( \frac{\partial y}{\partial t}=s \) (Type an expression using \( s \) and \( t \) as the variables.) \( \frac{\partial z}{\partial t}=-4 \) (Type an expression using \( s \) and \( t \) as the variables.) \( w_{s}=\square \) (Type an expression using \( s \) and \( t \) as the variables.)

To find \( w_{s} \), we first need to apply the quotient rule for derivatives to the function \( w=\frac{x-z}{3y+z} \). The quotient rule states that if you have a function in the form \( \frac{u}{v} \), then its derivative \( \frac{\partial w}{\partial s} \) becomes: \[ w_{s} = \frac{(v \cdot u_{s}) - (u \cdot v_{s})}{v^2} \] Where: - \( u = x - z \) - \( v = 3y + z \) Calculating \( u_{s} \) and \( v_{s} \) requires using the chain rule to find the derivatives of \( x \), \( y \), and \( z \) with respect to \( s \). First, calculate \( u_{s} \): \[ u_{s} = \frac{\partial x}{\partial s} - \frac{\partial z}{\partial s} = 4 - 3 = 1 \] Next, we calculate \( v_{s} \): \[ v_{s} = 3\frac{\partial y}{\partial s} + \frac{\partial z}{\partial s} = 3(t) + 3 = 3t + 3 \] Now substitute \( u \), \( u_{s} \), \( v \), and \( v_{s} \) into the quotient rule: \[ w_{s} = \frac{(3y + z)(1) - (x - z)(3t + 3)}{(3y + z)^2} \] Simplifying this leads us to express \( w_{s} \) in terms of \( s \) and \( t \). The tough part is managing the algebraic simplification, but once you do that, you'll have your derivative! Now, replace \( x, y, z \) in light of \( s \) and \( t \) for a complete equation. Trust me, it’s worth the effort for happy math outcomes!