91Ó°ÊÓ

Differentiation is a fundamental concept in calculus. It involves finding the derivative of a function, which measures how the function's output changes as its input changes. In simpler terms, the derivative tells you the rate at which one quantity changes with respect to another. In this exercise, we need to find the derivative of the given function, which will tell us how the function behaves as the input, or the variable, changes.

When you differentiate a function, you're essentially calculating the slope of the tangent line at any given point along the curve represented by that function. This slope provides a great deal of insight into the function's behavior, such as whether it's increasing or decreasing, and by how much, in different sections of its graph.

The power rule is a quick and efficient method for finding the derivative of a term that is in the form of a power. Specifically, if you have a term like \(x^n\), the power rule tells you that the derivative is \(nx^{n-1}\). It's a handy shortcut for differentiation, especially when dealing with simple polynomials.

In our exercise, the term \(5x^{-1}\) was differentiated using the power rule. To apply it, we multiplied the power by the coefficient, resulting in \(-5\), and then decreased the exponent by one to get \(-2\). Therefore, the derivative of \(5x^{-1}\) becomes \(-5x^{-2}\). Mastering the power rule can greatly simplify calculus problems by reducing complex differentiation to a series of simple steps.

Simplification involves reducing an expression to its simplest form. This practice is crucial because it makes mathematical operations more manageable and clearer to work with. In the given exercise, simplification was an essential step before applying differentiation.

Initially, we had the expression \(x^2\left( \frac{2}{x^2} + \frac{5}{x^3} \right)\), which looks complex. By distributing \(x^2\) and then simplifying each term—specifically \(x^2 \cdot \frac{2}{x^2}\) to \(2\), and \(x^2 \cdot \frac{5}{x^3}\) to \(\frac{5}{x}\)—the expression became much simpler: \(2 + 5x^{-1}\). This simplification step recasts the problem into a much easier form for differentiation.

A function is a relation that uniquely associates an input with a corresponding output. In mathematics, it is a rule that assigns to each element in a set exactly one element in another set. Functions are often represented as equations, like the one in this exercise: \(f(x) = x^2\left( \frac{2}{x^2} + \frac{5}{x^3} \right)\).

Understanding the properties and behavior of a function is key when working with calculus. Knowing how a function changes (the rate of change) involves studying its derivative. In our exercise, we analyzed and simplified the function, and then derived it, to understand how it changes with respect to \(x\). This understanding is foundational for solving more complex problems in calculus and applied mathematics.