Question

Use the limit definition of partial derivatives to find \(f_{x}(x, y)\) and \(f_{y}(x, y)\). \(f(x, y)=2 x+3 y\)

Short answer

The first order partial derivatives of the function \(f(x, y)=2 x+3 y\) are \(f_{x}(x, y)=2\) and \(f_{y}(x, y)=3\)

Step-by-step solution

Finding the partial derivative with respect to x: \(f_{x}(x, y)\)

Let's use the limit definition and replace \(f(x, y)\) with \(2x+3y\). We get \(\lim_{h \to 0} \frac{(2(x+h)+3y)-(2x+3y)}{h}\). Simplifying this gives to \(\lim_{h \to 0} \frac{2h}{h}\). Taking the limit as h approaches 0 gives \(f_{x}(x, y)=2\)

Finding the partial derivative with respect to y: \(f_{y}(x, y)\)

Similarly, we use the limit definition and replace \(f(x, y)\) with \(2x+3y\). This time we get \(\lim_{j \to 0} \frac{(2x+3(y+j))-(2x+3y)}{j}\). Simplifying this gives \(\lim_{j \to 0} \frac{3j}{j}\). Taking the limit as j approaches 0 gives \(f_{y}(x, y)=3\)

Key concepts

Limit Definition of Partial Derivatives

The limit definition of partial derivatives is an essential concept in multivariable calculus that extends the idea of a derivative to functions of several variables. When dealing with functions like f(x, y)=2x+3y, the partial derivative with respect to one variable, say x, is found by taking the limit of the difference quotient as h approaches zero, but treating all other variables as constants.

In mathematical terms, to find the partial derivative of f with respect to x, denoted as \(f_{x}(x, y)\), we compute \(\lim_{h \to 0} \frac{f(x+h, y) - f(x, y)}{h}\). For the given function, this limit simplifies to \(\lim_{h \to 0} \frac{2h}{h}\) which results in \(2\), signifying that the slope of the function f in the direction of the x-axis is constant.

This concept is crucial in understanding how a function changes in one direction while keeping all other variables constant.

Multivariable Calculus

Multivariable calculus involves the study of calculus beyond one-dimensional space, extending into multiple dimensions, typically \(\mathbb{R}^{n}\) where \(n > 1\). It deals with functions that have more than one independent variable, such as \(f(x, y)\) from our exercise. In terms of partial derivatives, multivariable calculus helps us understand how a function behaves with respect to each variable independently.

Understanding Functional Relationships

With functions of multiple variables, you cannot simply plot them on a two-dimensional graph as you would for a single variable function. Instead, you can visualize each partial derivative graphically as a slice of the multidimensional surface that represents the entire function. The concept of holding variables constant becomes a powerful tool to analyze these complex surfaces and understand their slopes and rates of change.

Differentiation

In mathematics, differentiation is the process of finding the derivative of a function. It measures how a function's value changes as its input changes. The derivative of a single variable function is a straightforward application, but with multivariable functions, we consider partial derivatives.

The process of finding partial derivatives is known as partial differentiation. As seen in our exercise solution, \(f_{x}(x, y)=2\) and \(f_{y}(x, y)=3\), show that the rate of change of the function \(f\) with respect to \(x\) and \(y\) is constant. Differentiation as a concept goes beyond mechanical calculation; it is about understanding the rate at which quantities change and applying this understanding to model real-world scenarios like optimizing functions for minimum or maximum values, which is an essential skill in fields such as physics, engineering, and economics.