Question

Let \(w=f(x, y)\) be a function where \(x\) and \(y\) are functions of a single variable \(t\). Give the Chain Rule for finding \(d w / d t\)

Short answer

The chain rule is given by: \(dw/dt = (∂w/∂x) * dx/dt + (∂w/∂y) * dy/dt\).

Step-by-step solution

Recall the Chain Rule

The chain rule states that the derivative of a composite function is the derivative of the exterior function times the derivative of the interior function. In terms of the variables given, the chain rule can be formulated as follows: \(dw/dt = (∂w/∂x) * dx/dt + (∂w/∂y) * dy/dt\). In other words, the derivative of \(w\) with respect to \(t\) is equal to the partial derivative of \(w\) with respect to \(x\) times the derivative of \(x\) with respect to \(t\), plus the partial derivative of \(w\) with respect to \(y\) times the derivative of \(y\) with respect to \(t\).

Key concepts

Partial Derivatives

When dealing with functions of multiple variables, partial derivatives come into play. Unlike the derivative of a function of a single variable, which measures how the function changes as its input changes, a partial derivative measures how a function changes as one of its several variables changes, while keeping the other variables constant.

For instance, let's consider a function with two variables, such as temperature over a geographical area, where the function is defined as T(x, y), with x and y representing longitude and latitude, respectively. The partial derivative of T with respect to x, denoted as \(\partial T / \partial x\), calculates how temperature changes as you move east or west (longitude), while holding the latitude constant. Conversely, \(\partial T / \partial y\) measures the changes in temperature moving north or south (latitude), with the longitude held constant.

In the context of the chain rule, these partial derivatives serve an integral role in calculating the rate of change of a function with respect to an intermediary variable, as illustrated in the textbook example with function w.

Composite Functions

A composite function occurs when one function is embedded within another function. Think of it like a nesting doll, where we have an outer function and an inner function, each affecting the final output. In mathematical form, if we have two functions, say g(t) and f(x), and x itself is a function of t, then f(g(t)) is a composite function.

Imagine you're inside a funhouse where every mirror distorts your reflection in a unique way. If you stand between two mirrors, the reflection of the first mirror becomes the 'input' for the second mirror. This mirrors the idea of composite functions, where the output of the first function serves as the input for the second.

To find the derivative of a composite function, we need to apply the chain rule - the cornerstone concept which establishes the relationship between the derivatives of these functions. The chain rule lets us unpack the composite function into its parts, allowing us to calculate the rate of change of the entire system in a step-by-step manner.

Derivative of a Function

The derivative of a function is a fundamental concept in calculus that describes the rate of change of a function's output with respect to its input. In simpler terms, it tells us how fast the value of the function is changing at any point. For a function y = f(x), the derivative, denoted as \(f'(x)\) or \(\frac{dy}{dx}\), provides the slope of the tangent line to the function's graph at any point.

For example, if the function represents the distance traveled over time, the derivative would give the velocity, or the rate of distance covered, at any given moment. This is akin to looking at the speedometer of your car while driving - it shows your speed at the precise moment, not the average speed over your trip.

Understanding the derivative is crucial for applying the chain rule effectively since the chain rule is essentially about finding the derivative of a composite function. Each step in the chain rule represents the application of the derivative to different parts of the composition, helping us to understand complex dynamic systems.