91Ó°ÊÓ

Partial derivatives are a fundamental concept in calculus, especially for functions with more than one variable. They allow us to understand how a function changes as each of its inputs is varied, while keeping all other inputs constant. This is akin to zooming in on the function behavior with respect to one variable at a time.

In the context of the exercise, where we are examining functions like \( z=f(x, y) \), the partial derivative \( \frac{\partial z}{\partial t} \) shows how \( z \) changes with respect to \( t \), while keeping \( s \) constant. Similarly, \( \frac{\partial z}{\partial s} \) indicates the change in \( z \) when \( s \) varies and \( t \) remains unchanged.

Here are some key points to remember about partial derivatives:

• They are represented with the partial symbol \( \partial \), unlike ordinary derivatives which use \( d \).
• To find a partial derivative, treat all other variables as constants.

Multivariable functions are functions with more than one input variable. These functions are common in many fields such as physics, engineering, and economics where multiple factors affect the outcome. For example, in the exercise above, \( z = f(x, y) \) is a function of two variables, \( x \) and \( y \), which themselves are functions of \( t \) and \( s \).

Working with multivariable functions involves:

• Understanding how each input variable affects the output.
• Using tools like partial derivatives to analyze these effects individually.
• Applying the Chain Rule to explore the connection between changing inputs and the resultant change in the function's output.

These behaviors make calculating derivatives and understanding their impact crucial, especially when applying the chain rule to derive relationships like \( \frac{\partial z}{\partial t} \) and \( \frac{\partial z}{\partial s} \) in the exercise.

Function composition involves forming a new function by applying one function to the result of another. It is a powerful concept in calculus used to simplify the process of differentiation when multiple functions depend on one another, as seen in our exercise.

In the given exercise, we have three functions composed together:

• \( z = f(x, y) \), which is the main function depending on two intermediate variables.
• \( x = g(t, s) \) and \( y = h(t, s) \), which are dependent on time \( t \) and another variable \( s \).

Function composition is crucial here because the value of \( z \) changes not just directly with \( x \) and \( y \), but indirectly with how \( t \) and \( s \) influence \( x \) and \( y \).

This layered relationship is why the chain rule is used in differentiation; by applying the chain rule, we break down the effect of \( t \) and \( s \) on \( z \) in manageable parts, isolating each influence on the outcome, which is precisely what solving \( \frac{\partial z}{\partial t} \) and \( \frac{\partial z}{\partial s} \) accomplishes.