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Differentiation is a fundamental concept in calculus that involves finding the rate at which a function is changing. When we talk about differentiation with respect to a variable, we are essentially calculating the derivative of that function. The derivative gives us a new function that tells us the slope of the tangent line to the curve at any given point.

For a function given as \( f(x) \), its derivative is often denoted as \( f'(x) \) or \( \frac{df}{dx} \). This process is indispensable when dealing with functions as it provides insight into the behavior of a function, such as finding maximum or minimum points, understanding the function's increasing or decreasing intervals, and modeling real-world situations accurately.

In the exercise, we calculated the derivative of a polynomial function term by term. Each term's derivative was found to provide the rate of change of each independent term, which then combined to express the overall rate of change of the function.

The power rule is a mighty tool used in differentiation. It simplifies the process of finding the derivative of a function. It applies to functions of the form \( x^n \), where \( n \) is any real number. The rule is stated as: if \( f(x) = x^n \), then the derivative \( f'(x) = nx^{n-1} \).

This means you bring down the exponent as a coefficient and then reduce the power by one. This simple rule speeds up differentiation significantly, especially with polynomial expressions.

In the example given, we applied the power rule for each term separately:

• For \( x^2 \), derivative is \( 2x^{2-1} = 2x \).
• For \( -3x \), since it can be seen as \( -3x^1 \), the derivative is simply \( -3 \).
• For \( -3x^{-2} \), the derivative becomes \( 6x^{-3} \) as per the power rule.

By successfully applying the power rule, you can effortlessly differentiate polynomial and even non-standard power functions.

Functions are the backbone of calculus and mathematics as a whole. A function links an input to a single output through a specified rule. Typically, functions are represented as \( f(x) \), where \( x \) is the variable input, and \( f(x) \) represents the output. Functions can take many forms, including linear, polynomial, exponential, and trigonometric, among others.

Understanding the behavior of a function begins with knowing its domain and range, how it increases or decreases, and whether it has any turning points or points of inflection. Derivatives are a great help in analyzing functions because they supply information on the rate of change.

In the exercise, the function given was polynomial: \( f(x) = x^2 - 3x - 3x^{-2} \). Each term in the function had a distinct behavior based on its degree and coefficient. Differentiating each term separately gives insight into how each component of the function contributes to its overall rate of change.