A and B are both functions of two variables x and y, and A /...

$A$ and $B$ are both functions of two variables $x$ and $y$, and $A / B=C$. Show that $$ \left.\frac{\partial x}{\partial y}\right|_C=\frac{\left.\frac{\partial \ln B}{\partial y}\right|_x-\left.\frac{\partial \ln A}{\partial y}\right|_x}{\left.\frac{\partial \ln A}{\partial x}\right|_y-\left.\frac{\partial \ln B}{\partial x}\right|_y} $$ [Hint: develop the left-hand side, and don't forget that for any function $f,(\mathrm{~d} / \mathrm{d} x)(\ln f)=(1 / f) \mathrm{d} f / \mathrm{d} x]$.

$A$ and $B$ are both functions of two variables $x$ and $y$, and $A / B=C$. Show that
$$
\left.\frac{\partial x}{\partial y}\right|_C=\frac{\left.\frac{\partial \ln B}{\partial y}\right|_x-\left.\frac{\partial \ln A}{\partial y}\right|_x}{\left.\frac{\partial \ln A}{\partial x}\right|_y-\left.\frac{\partial \ln B}{\partial x}\right|_y}
$$
[Hint: develop the left-hand side, and don't forget that for any function $f,(\mathrm{~d} / \mathrm{d} x)(\ln f)=(1 / f) \mathrm{d} f / \mathrm{d} x]$.

Key Concepts

Chain Rule

The chain rule is a fundamental differentiation tool used to calculate the derivative of composite functions. In the context of functions with multiple variables, it provides a method for differentiating a function with several interdependent variables by accounting for the layers of dependency between the variables.

Logarithmic Differentiation

Logarithmic differentiation is a technique that simplifies the process of differentiating complicated functions by taking the logarithm of both sides of an equation. This method is particularly useful when dealing with products, quotients, or variable exponents, as it transforms multiplicative relationships into additive ones, making differentiation more manageable.

Total Derivatives

The total derivative accounts for the way a function changes with respect to a variable that affects several independent variables simultaneously. It combines the contributions of all the partial derivatives of the function, making it indispensable when evaluating the derivative along curves or surfaces defined implicitly by conditions such as a constant value of another function.

Implicit Differentiation

Implicit differentiation is a technique used to differentiate functions that are defined implicitly rather than explicitly. In problems where a relationship between variables is given without explicitly solving for one variable in terms of the others, implicit differentiation allows one to compute derivatives by differentiating both sides of the equation while treating some variables as functions of others.

Partial Derivatives

Partial derivatives measure the rate of change of a function with respect to one of its variables when all other variables are held constant. They are crucial in settings where functions depend on more than one variable and are used to analyze the sensitivity of the function to changes in each independent variable separately.

Functions of Multiple Variables

This concept involves understanding that functions can depend on several independent variables. In such cases, the variation of a function with respect to one variable is studied while keeping the other variables fixed. This provides the foundation for understanding how changes in one variable affect the function's output in a multidimensional space.

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