91Ó°ÊÓ

When faced with a function in the form of a quotient—a ratio of two functions—one useful tool for differentiation is the quotient rule. It is essential in single-variable calculus, and just as much in multivariable calculus. The quotient rule states that the derivative of \( \frac{u(v)}{v(w, z)} \) with respect to a variable, say \( w \), is given by \( \frac{v(w, z)\frac{d u}{d w} - u(v)\frac{d v}{d w}}{[v(w, z)]^2} \).
Understanding the quotient rule allows us to tackle the differentiation of rational functions, where the numerator and the denominator are both functions of the variable with respect to which we're differentiating. It's important to apply the rule correctly by differentiating the numerator and the denominator separately and then combining the results as per the formula.

Multivariable calculus extends the concepts of single-variable calculus to functions of several variables. In fields such as physics, engineering, and economics, variables often depend on multiple factors, which is where multivariable calculus becomes essential. Instead of dealing with a single input and output, we consider functions with inputs like \( w \) and \( z \), and how they change with respect to each variable. Concepts such as partial derivatives, gradient vectors, and multiple integrals are fundamental in studying the surfaces, solids, and change rates in multivariable systems.

Differentiation is the process of finding the derivative of a function, which represents the rate at which the function's value is changing at a particular point. In the context of multivariable calculus, differentiation is performed with respect to each variable independently while holding the other variables constant. This gives rise to partial derivatives, which represent the rate of change of a multivariable function with respect to one variable at a time.

A partial derivative with respect to a variable, such as \( w \), is denoted as \( \frac{\partial}{\partial w} \) and is calculated by differentiating the function concerning \( w \) while treating all other variables as constants. In our exercise, the differentiation of \( \frac{w}{w^2 + z^2} \) with respect to \( w \) involves using the quotient rule, treating \( z \) as a constant. This results in \( \frac{\partial}{\partial w} f(w, z) = \frac{-w^2+z^2}{(w^2+z^2)^2} \) after substituting the differentiated terms and simplifying. It tells us how the function \( f \) changes as \( w \) varies, with \( z \) fixed.

Similarly, the partial derivative with respect to \( z \) is represented as \( \frac{\partial}{\partial z} \) and finds the rate of change of the function when \( z \) varies and \( w \) is held constant. Applying the quotient rule in our example gives \( \frac{\partial}{\partial z} f(w, z) = \frac{-2wz}{(w^2+z^2)^2} \) after differentiating the numerator where \( w \) is considered constant and \( z \) is the variable of interest. This partial derivative highlights how changes in \( z \) affect the value of our function \( f \) while \( w \) remains unchanged.