91Ó°ÊÓ

The quotient rule is a method used in calculus to find the derivative of a function that is the ratio of two differentiable functions. Given two functions, say \(u(x)\) and \(v(x)\), the quotient rule states:
\[ \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \]
This might look complex at first, but it's about applying the product rule in a structured way for the numerator while keeping the denominator squared. Here, the function we dealt with was:
\[ f(x, y) = \frac{x - y}{x + y} \]
When we used the quotient rule, we treated the numerator \(u = x - y\) and the denominator \(v = x + y\) as separate functions. Applying this method carefully gives us the partial derivatives as required.

Multivariable calculus extends the principles of calculus to functions of more than one variable, such as \(f(x, y)\). Instead of dealing with single-variable functions, we now consider how a function changes as multiple variables change simultaneously. This requires finding partial derivatives. In this example, our function depends on both \(x\) and \(y\). Understanding how each variable independently affects the function is key to analyzing its behavior and properties. We used tools from multivariable calculus to break down and handle each part of the function separately, analyzing changes with respect to \(x\) and \(y\) independently.

Partial differentiation is the process of finding the derivative of a function with respect to one variable while treating other variables as constants. In simpler terms, it means focusing on changes in one direction at a time. For instance, finding \( \frac{\partial f}{\partial x} \) means looking at how \( f \) changes as \( x \) changes while keeping \( y \) fixed. To compute the partial derivatives of \( f(x, y) = \frac{x - y}{x + y} \), we:

• Treated \( y \) as a constant to find \( \frac{\partial f}{\partial x} \)
• Treated \( x \) as a constant to find \( \frac{\partial f}{\partial y} \)

By treating one variable as fixed and using the quotient rule, we successfully found:
\[ \frac{\partial f}{\partial x} = \frac{2y}{(x + y)^2} \] and
\[ \frac{\partial f}{\partial y} = \frac{-2x}{(x + y)^2} \]