Implicit Differentiation and Partial Derivatives

We are given an implicit function $G(x, y, z) = 0$. We need to find the value of the product $\left(\frac{\partial z}{\partial x} \times \frac{\partial x}{\partial y} \times \frac{\partial y}{\partial z}\right)$. We assume $G$'s partial derivatives exist and are non-zero where needed.

Finding Partial Derivatives

Using the rules of implicit differentiation:

• To find $\frac{\partial z}{\partial x}$, we treat $y$ as a constant and differentiate $G$ with respect to $x$, considering $z$ as a function of $x$ and $y$:

  $ \frac{\partial G}{\partial x} + \frac{\partial G}{\partial z} \frac{\partial z}{\partial x} = 0 $

  Solving for $\frac{\partial z}{\partial x}$ yields:

  $ \frac{\partial z}{\partial x} = -\frac{\frac{\partial G}{\partial x}}{\frac{\partial G}{\partial z}} $

• To find $\frac{\partial x}{\partial y}$, we treat $z$ as a constant and differentiate $G$ with respect to $y$, considering $x$ as a function of $y$ and $z$:

  $ \frac{\partial G}{\partial y} + \frac{\partial G}{\partial x} \frac{\partial x}{\partial y} = 0 $

  Solving for $\frac{\partial x}{\partial y}$ yields:

  $ \frac{\partial x}{\partial y} = -\frac{\frac{\partial G}{\partial y}}{\frac{\partial G}{\partial x}} $

• To find $\frac{\partial y}{\partial z}$, we treat $x$ as a constant and differentiate $G$ with respect to $z$, considering $y$ as a function of $x$ and $z$:

  $ \frac{\partial G}{\partial z} + \frac{\partial G}{\partial y} \frac{\partial y}{\partial z} = 0 $

  Solving for $\frac{\partial y}{\partial z}$ yields:

  $ \frac{\partial y}{\partial z} = -\frac{\frac{\partial G}{\partial z}}{\frac{\partial G}{\partial y}} $

Calculating the Product

Now, we multiply these three partial derivatives:

$ \left(\frac{\partial z}{\partial x} \times \frac{\partial x}{\partial y} \times \frac{\partial y}{\partial z}\right) = \left(-\frac{\frac{\partial G}{\partial x}}{\frac{\partial G}{\partial z}}\right) \times \left(-\frac{\frac{\partial G}{\partial y}}{\frac{\partial G}{\partial x}}\right) \times \left(-\frac{\frac{\partial G}{\partial z}}{\frac{\partial G}{\partial y}}\right) $

Simplify the expression by cancelling terms:

$ = (-1) \times (-1) \times (-1) \times \left(\frac{\frac{\partial G}{\partial x}}{\frac{\partial G}{\partial x}}\right) \times \left(\frac{\frac{\partial G}{\partial y}}{\frac{\partial G}{\partial y}}\right) \times \left(\frac{\frac{\partial G}{\partial z}}{\frac{\partial G}{\partial z}}\right) $

$ = (-1) \times 1 \times 1 \times 1 $

$ = -1 $

Final Result

The value of the product $\left(\frac{\partial z}{\partial x} \times \frac{\partial x}{\partial y} \times \frac{\partial y}{\partial z}\right)$ is -1.