Question

Find \(\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}\). $$ f(x)=\frac{4}{x} $$

Short answer

The derivative of \(f(x) = \frac{4}{x}\) according to the definition of a derivative, is \(\frac{-4}{x^2}\)

Step-by-step solution

Substitute the Function

Start by substituting \(f(x+\Delta x)\) and \(f(x)\) into the equation using \(f(x) = \frac{4}{x}\): \(\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x} = \lim _{\Delta x \rightarrow 0} \frac{\frac{4}{x + \Delta x} - \frac{4}{x}}{\Delta x}\)

Find Common Denominator

Find a common denominator for the two fractions in the numerator, which would be \(x(x+\Delta x)\) and simplify: \(\lim _{\Delta x \rightarrow 0} \frac{\frac{4x}{x(x+\Delta x)} - \frac{4(x+\Delta x)}{x(x+\Delta x)}}{\Delta x} = \lim _{\Delta x \rightarrow 0} \frac{4x - 4x - 4\Delta x}{x(x+\Delta x)\Delta x}\)

Simplify the expression

Simplify the expression further by canceling the \(4x - 4x\) and removing common factor in numerator and denominator: \(\lim _{\Delta x \rightarrow 0} \frac{- 4}{x(x+\Delta x)}\)

Take the limit

Now, take the limit as \(\Delta x\) approaches 0, resulting in: \(\lim _{\Delta x \rightarrow 0} \frac{-4}{x(x+\Delta x)} = \frac{-4}{x^2}\)

Key concepts

Calculus

Calculus is a branch of mathematics that studies the rates of change and quantities that accrue over time. It is divided into two major areas: differential calculus, focusing on the rate at which quantities change, and integral calculus, dealing with the accumulation of quantities.

In the context of the exercise provided, where we deal with the limit of a function as \(\Delta x\) approaches zero, we are venturing into differential calculus. This particular limit is foundational to calculus as it leads to the concept of the derivative, which measures how a function changes as its input changes.

Understanding how to handle a limit is crucial for students because it sets the groundwork for further studies in calculus such as derivatives, integrals, and their applications in various fields including physics, engineering, and economics.

Continuity and Limits

The concept of continuity and limits in calculus is one of the most fundamental ideas. A function is said to be continuous at a point if the function's value at that point is equal to the limit of the function as it approaches that point. In other words, there are no jumps, holes, or breaks in the function at that point.

In the provided exercise, we are asked to find the limit of a function as \(\Delta x\) approaches 0. This calculation helps determine the behavior of the function around a particular point and is a crucial aspect of ensuring a function is continuous.

An understanding of limits is also essential for learning about the concept of derivatives. When working with limits, especially when variables tend to zero, students are encouraged to familiarize themselves with limit laws and the concept of the infinitesimal, which can often help in simplifying complex expressions before evaluating the limit.

Derivative Definition

The derivative of a function at a certain point provides the rate at which the function value changes with respect to change in its input value. In its most basic definition, the derivative of a function is the limit of the ratio of the change in the function's output to the change in the function's input as this change in input gets infinitely small.

From the exercise, we are effectively working on finding the derivative of the function \(f(x)=\frac{4}{x}\) at any point \(x\) using the definition of the derivative. We calculate the derivative by evaluating the limit of the difference quotient as \(\Delta x\) approaches zero, which is an application of the fundamental principle of derivatives. This process highlights how studying limits is directly tied to understanding the concept of derivatives in calculus.

Learning to rigorously compute derivatives from their definitions not only helps students to grasp how derivatives are derived but also empowers them with the foundational skills necessary for more advanced calculus topics, such as optimization and differential equations.