We are given:

$u = 4x + 2y, \quad v = 2x - 4y$

We need to find the value of the following product:

$\left(\frac{\partial u}{\partial x}\right)_v \cdot \left(\frac{\partial x}{\partial u}\right)_v \cdot \left(\frac{\partial v}{\partial y}\right)_u \cdot \left(\frac{\partial y}{\partial v}\right)_x$

Step 1: Compute partial derivatives

1. $\frac{\partial u}{\partial x}$: $u = 4x + 2y \quad \Rightarrow \quad \frac{\partial u}{\partial x} = 4$

2. $\frac{\partial v}{\partial y}$: $v = 2x - 4y \quad \Rightarrow \quad \frac{\partial v}{\partial y} = -4$

Step 2: Express $\frac{\partial x}{\partial u}$ and $\frac{\partial y}{\partial v}$ using implicit differentiation.

We need to express $x$ and $y$ in terms of $u$ and $v$ from the given system of equations:

$u = 4x + 2y \quad \text{and} \quad v = 2x - 4y$

Solve these equations simultaneously.

From the second equation, express $x$ in terms of $v$ and $y$: $v = 2x - 4y \quad \Rightarrow \quad x = \frac{v + 4y}{2}$

Substitute this expression for $x$ into the first equation: $u = 4\left(\frac{v + 4y}{2}\right) + 2y = 2(v + 4y) + 2y = 2v + 8y + 2y = 2v + 10y$ Thus, we have: $u = 2v + 10y \quad \Rightarrow \quad y = \frac{u - 2v}{10}$

Now substitute this $y$ back into the expression for $x$: $x = \frac{v + 4\left(\frac{u - 2v}{10}\right)}{2} = \frac{v + \frac{4u - 8v}{10}}{2} = \frac{5v + 4u - 8v}{10} = \frac{4u - 3v}{10}$

Step 3: Calculate remaining partial derivatives

Now that we have $x$ and $y$ in terms of $u$ and $v$, compute the derivatives:

3. $\frac{\partial x}{\partial u}$: $x = \frac{4u - 3v}{10} \quad \Rightarrow \quad \frac{\partial x}{\partial u} = \frac{4}{10} = \frac{2}{5}$

4. $\frac{\partial y}{\partial v}$: $y = \frac{u - 2v}{10} \quad \Rightarrow \quad \frac{\partial y}{\partial v} = -\frac{2}{10} = -\frac{1}{5}$

Step 4: Multiply all derivatives

The product of the partial derivatives is: