Question

Let \(w=f(x, y)\) be a function where \(x\) and \(y\) are functions of two variables \(s\) and \(t\). Give the Chain Rule for finding \(\partial w / \partial s\) and \(\partial w / \partial t\)

Short answer

The Chain Rule for finding the partial derivatives \(\partial w / \partial s\) and \(\partial w / \partial t\) are \(\partial w / \partial s = (\partial f/ \partial x) \cdot (\partial x/ \partial s) + (\partial f/ \partial y) \cdot (\partial y/ \partial s)\) and \(\partial w / \partial t = (\partial f/ \partial x) \cdot (\partial x/ \partial t) + (\partial f/ \partial y) \cdot (\partial y/ \partial t)\), respectively.

Step-by-step solution

Apply the Chain Rule for \(\partial w / \partial s\)

Starting with \(\partial w / \partial s\), we look to the chain rule for a composite function which states: if \(z = f(g(x))\), then \(\partial z/ \partial x = f'(g(x)) \cdot g'(x)\). Applying this concept to the given function, we have \(w = f(x(s, t), y(s, t))\), therefore \(\partial w / \partial s = (\partial f/ \partial x) \cdot (\partial x/ \partial s) + (\partial f/ \partial y) \cdot (\partial y/ \partial s)\).

Apply the Chain Rule for \(\partial w / \partial t\)

We will do the same for \(\partial w / \partial t\), applying the chain rule again, we have: \(\partial w / \partial t = (\partial f/ \partial x) \cdot (\partial x/ \partial t) + (\partial f/ \partial y) \cdot (\partial y/ \partial t)\).

Generalize the Results

So, the general results for the chain rule for several variables when given a function \(w = f(x, y)\), where \(x\) and \(y\) are functions of \(s\) and \(t\), is \(\partial w / \partial s = (\partial f/ \partial x) \cdot (\partial x/ \partial s) + (\partial f/ \partial y) \cdot (\partial y/ \partial s)\) and \(\partial w / \partial t = (\partial f/ \partial x) \cdot (\partial x/ \partial t) + (\partial f/ \partial y) \cdot (\partial y/ \partial t)\). This represents the chain rule for partial derivatives.

Key concepts

Partial Derivatives

When we dive into the world of multivariable calculus, one of the essential tools we encounter are partial derivatives. Partial derivatives measure how a function changes as only one variable changes while keeping other variables constant. Imagine you're at a mountain peak and you're able to move either north or upward. Moving north changes your horizontal position, similar to changing one variable in a function, while your altitude stays the same. Similarly, moving upward changes your altitude without affecting your north-south position. This is what taking a partial derivative feels like; it's like examining the slope of the terrain in a single direction.

For a function like \(w = f(x, y)\), where \(w\) depends on two variables, \(x\) and \(y\), the partial derivative with respect to \(x\), denoted as \(\partial w / \partial x\), looks at how \(w\) changes when \(x\) varies but \(y\) remains fixed. And vice versa for \(\partial w / \partial y\). Understanding partial derivatives is crucial because they are the building blocks for more complex calculations in multivariable calculus, like gradients and Jacobian matrices.

Multivariable Calculus

Multivariable calculus extends the concepts of single-variable calculus to functions with multiple inputs. Imagine you're a chef trying to perfect a complex recipe that requires adjusting various ingredients to achieve the desired taste. Likewise, in multivariable calculus, we analyze how changing multiple ingredients (variables) simultaneously affects the overall dish (the function).

This field delves into topics like partial derivatives, multiple integrals, and vector calculus. The Chain Rule for partial derivatives, for example, is a fundamental theorem that helps us deal with functions within functions (called composite functions), where each component may depend on multiple variables. It's like tweaking several spices in our recipe at once and observing the intricate ways in which the flavors combine and evolve. Understanding how these variables interact is essential for solving real-world problems in physics, engineering, and economics, which often depend on many fluctuating factors.

Composite Functions

Composite functions come into play when we have a function inside another function—imagine a Russian nesting doll, where each layer encompasses another. In the world of calculus, this is similar to having a situation where one variable is a function of other variables, which then feed into a second function. It's like setting your GPS for a road trip; your location changes with time, and these changes influence your arrival time, which is a function of your location.

In mathematical terms, if you have \(w = f(x, y)\), and both \(x\) and \(y\) are themselves functions of other variables (say, \(s\) and \(t\)), we're dealing with a composite function. To understand how changes in the underlying variables (\(s\) and \(t\)) affect the output \(w\), we use the Chain Rule for partial derivatives. It's a systematic way to unravel the influence of each nested variable. It ensures that when assessing the impact of a long road trip on your arrival, you consider both your speed and the potential traffic along the way.