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Differential calculus is a fundamental branch of mathematics that focuses on the concept of change. It provides the tools we need to understand how functions change at any point. This is particularly useful in finding the rate at which something happens, like speed or acceleration.
In the context of functions, differential calculus helps us determine the slope of a curve. The slope can tell us how steeply the function is increasing or decreasing at a particular point. This is achieved by finding the derivative of the function. Critical points, which are places where the slope is zero or undefined, are crucial in understanding the behavior of the function, such as identifying maxima, minima, or points of inflection.

The derivative of a function measures how the function's value changes as its input changes. It is denoted as \(f'(x)\) for a function \(f(x)\). Finding derivatives is a way to understand the function's behavior, conditions of increasing or decreasing trends, and to locate critical points where significant changes in trends occur, such as maxima or minima.
To calculate a derivative, we apply rules like the power rule and product rule, depending on the structure of the function. In differential calculus, derivatives are essential as they provide a precise mathematical way to analyze changes, helping us to solve various real-world problems like optimization and motion analysis.

The power rule is a simple and efficient method used in calculus to find the derivative of functions of the form \(x^n\), where \(n\) can be any real number. It states that if \(f(x) = x^n\), then the derivative \(f'(x) = nx^{n-1}\). This rule is very helpful because it allows us to quickly calculate derivatives without needing to use more complicated methods.
For example, in the exercise, to find the derivative of \(x^{4/5}\), we use the power rule to get \((4/5)x^{-1/5}\). It's an incredibly powerful tool that makes working with polynomial expressions straightforward. Remembering this rule saves time and simplifies complex calculations, making it a foundational concept in calculus.

The product rule is essential when taking derivatives of functions that are products of two or more functions. When you have a function \(f(x)\) that can be written as a product \(u(x) \cdot v(x)\), where both \(u\) and \(v\) are functions of \(x\), the derivative \(f'(x)\) is found using the product rule.
The product rule states: \(f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)\). This means you need to differentiate each function separately, first \(u(x)\) and then \(v(x)\), and then sum up the products of each derivative with the other function. This method was applied in the exercise with the functions \(x^{4/5}\) and \((x-4)^{2}\). Understanding the product rule is crucial because it allows us to tackle more complex functions that involve multiplication, making it easier to analyze any function's behavior.