Problem 31 - Calculus an Applied Approach

Question

find the higher-order derivative. $$ f^{\prime \prime \prime}(x)=(3 x-1) / x \quad f^{(4)}(x) $$

Step-by-Step Solution

Answer

The fourth derivative of the function is \( f^{(4)}(x)=\frac{1}{x^{2}} \).

Step 1: Rewriting the expression

Firstly, rewrite the function to obtain a simpler derivative. \( f^{\prime \prime \prime}(x)=(3 x-1) / x \) can be rewritten as \( f^{\prime \prime \prime}(x)=3 - \frac{1}{x} \).

Step 2: Differentiating the function

Next step is to differentiate the rewritten function. The derivative of a constant vanishes, and the derivative of \( \frac{1}{x} \) is \( -\frac{1}{x^{2}} \). Therefore, the fourth derivative \( f^{(4)}(x) \) is \( f^{(4)}(x)=0 - \left( -\frac{1}{x^{2}} \right)=\frac{1}{x^{2}} \).

Key Concepts

Differentiation

Imagine you're on a road trip and you're keeping track of how far you've traveled over time. Differentiation is like a mathematical tool that tells you your speed at any given moment—it measures how fast you're going, not just the distance you've covered. This concept lies at the very heart of calculus, which deals with changes.

In technical terms, differentiation is the process of finding the derivative of a function. When you differentiate a function, you're essentially finding a new function that gives the rate at which the original function is changing at every point. Think of it as a mathematical microscope that lets you zoom in on a graph and see exactly how steep or flat it is at any point.

For your exercise, the differentiation of the function led from the third derivative to the fourth, similar to finding a new, more detailed layer of how the function's rate of change is evolving.

Calculus

Calculus is like the Swiss Army knife of mathematics. It is a field that allows us to understand the dynamics of change and motion. There are two main branches: differential calculus (which is all about differentiation, as mentioned earlier) and integral calculus (which deals with areas and accumulations). The two balance each other out, like opposite sides of the same coin.

For students, mastering calculus is a gateway to understanding complex change in physics, economics, and biology, among countless other fields. It's not just about finding higher-order derivatives or calculating areas under curves; it's about grasolvinganimating and predicting the behavior of systems that change in intricate ways.

The exercise you're working on uses differential calculus to explore deeper levels of rate changes within a function.

Derivative of a Function

The derivative of a function represents the spine of calculus. Imagine you're drawing a tangent line to a curve on a graph at a single point. The slope of that line is the derivative of the function at that point—it shows you how the function is changing right there and then. If you move along the curve, the slope of these tangent lines might change, and that's exactly what the derivative tells you: how the function's rate of change varies along its domain.

When you calculated the fourth derivative in your exercise, you were essentially asking, 'How is the rate of change of the rate of change of the rate of change of my function changing?'. It's a mouthful, but it's a way to describe the behavior of something that's evolving in a complex way—beyond just speeding up or slowing down. The higher-order derivatives provide a nuanced view into the dynamics of change that a function can exhibit.

For students, understanding how to efficiently find these derivatives, particularly by simplifying expressions first (as done in your exercise), is fundamental to mastering calculus and a testament to the elegance of mathematical progression.