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.)

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To find the derivatives $$ w_s $$ and $$ w_t $$ where $$ w = \frac{x - z}{3y + z} $$, we will use the chain rule and the given expressions for $$ x $$, $$ y $$, and $$ z $$.

### Step 1: Identify the expressions
We have:
- $$ w = \frac{x - z}{3y + z} $$
- $$ x = 3s + 4t $$
- $$ y = st $$
- $$ z = 3s - 4t $$

### Step 2: Calculate the partial derivatives of $$ w $$

#### Partial Derivative with respect to $$ s $$ ($$ w_s $$)
Using the quotient rule:
$$
w_s = \frac{(3y + z)(\frac{\partial (x - z)}{\partial s}) - (x - z)(\frac{\partial (3y + z)}{\partial s})}{(3y + z)^2}
$$

1. Calculate $$ \frac{\partial (x - z)}{\partial s} $$:
   $$
   \frac{\partial (x - z)}{\partial s} = \frac{\partial x}{\partial s} - \frac{\partial z}{\partial s} = 3 - 3 = 0
   $$

2. Calculate $$ \frac{\partial (3y + z)}{\partial s} $$:
   $$
   \frac{\partial (3y + z)}{\partial s} = 3\frac{\partial y}{\partial s} + \frac{\partial z}{\partial s} = 3t + 3 = 3t + 3
   $$

Now substituting these into the expression for $$ w_s $$:
$$
w_s = \frac{(3y + z)(0) - (x - z)(3t + 3)}{(3y + z)^2} = \frac{-(x - z)(3t + 3)}{(3y + z)^2}
$$

#### Partial Derivative with respect to $$ t $$ ($$ w_t $$)
Using the quotient rule:
$$
w_t = \frac{(3y + z)(\frac{\partial (x - z)}{\partial t}) - (x - z)(\frac{\partial (3y + z)}{\partial t})}{(3y + z)^2}
$$

1. Calculate $$ \frac{\partial (x - z)}{\partial t} $$:
   $$
   \frac{\partial (x - z)}{\partial t} = \frac{\partial x}{\partial t} - \frac{\partial z}{\partial t} = 4 - (-4) = 4 + 4 = 8
   $$

2. Calculate $$ \frac{\partial (3y + z)}{\partial t} $$:
   $$
   \frac{\partial (3y + z)}{\partial t} = 3\frac{\partial y}{\partial t} + \frac{\partial z}{\partial t} = 3\frac{\partial y}{\partial t} - 4
   $$

Now substituting these into the expression for $$ w_t $$:
$$
w_t = \frac{(3y + z)(8) - (x - z)(3\frac{\partial y}{\partial t} - 4)}{(3y + z)^2}
$$

### Step 3: Find $$ \frac{\partial y}{\partial t} $$
From the expression for $$ y $$:
$$
y = st \implies \frac{\partial y}{\partial t} = s
$$

### Final Expressions
Thus, we have:
- $$ w_s = \frac{-(x - z)(3t + 3)}{(3y + z)^2} $$
- $$ w_t = \frac{(3y + z)(8) - (x - z)(3s - 4)}{(3y + z)^2} $$

### Summary of Derivatives
1. $$ \frac{\partial x}{\partial s} = 3 $$
2. $$ \frac{\partial y}{\partial s} = t $$
3. $$ \frac{\partial z}{\partial s} = 3 $$
4. $$ \frac{\partial x}{\partial t} = 4 $$
5. $$ \frac{\partial y}{\partial t} = s $$
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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