Question

What is the name of the rule which states that \(\frac{d}{d x}\left(x^{n}\right)=\) \(n x^{n-1},\) where \(n>0\) is an integer?

Short answer

The rule is called the "Power Rule".

Step-by-step solution

Recognize the Rule Format

The given expression \(\frac{d}{dx}(x^n) = nx^{n-1}\) is a differentiation rule in calculus for polynomial functions.

Identify the Power Rule

This rule specifically states that when differentiating a power function \(x^n\), you multiply the exponent \(n\) by the function and decrease the exponent by one. The rule is known as the "Power Rule" of differentiation.

Key concepts

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate at which a function is changing at any given point. It's like finding the slope of a tangent line to the curve of the function at a specific spot. Differentiation is essential in various fields including physics, engineering, and economics. To differentiate a function, such as a polynomial, you apply certain rules and methods. One of the most commonly used rules for differentiation is the Power Rule, especially useful for polynomial functions. In essence, differentiation allows us to understand how a small change in one variable affects another, making it a powerful tool for analyzing mathematical models.

Polynomial Functions

Polynomial functions consist of variables raised to whole number exponents and multiplied by coefficients. They have the general form:\[P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \]where \(a_n, a_{n-1}, \ldots, a_1, a_0\) are constants called coefficients, and \(n\) is a non-negative integer representing the degree of the polynomial. The degree indicates the highest power of the variable in the function.Polynomial functions can be straight lines, parabolas, or more complex curves, depending on their degree. They are smooth, continuous functions without breaks or gaps, making them simpler to analyze and understand across their domains. When differentiating polynomial functions, the process typically involves using the Power Rule, making it straightforward to determine the function’s behavior.

Exponent Rules

Exponent rules are key to understanding how variables work when raised to powers. Essential rules include:- **Product of Powers**: \(x^a \cdot x^b = x^{a+b}\)- **Power of a Power**: \((x^a)^b = x^{a\cdot b}\)- **Quotient of Powers**: \(\frac{x^a}{x^b} = x^{a-b}\)When using the Power Rule for differentiation, these exponent rules are invaluable. They help simplify expressions before finding derivatives. For instance, when differentiating functions like \(x^n\), the Power Rule directly uses the exponent on the variable. This makes understanding and applying fitting exponent rules crucial in calculus.These rules are not only foundational for differentiation but also for more complex operations such as integration and solving polynomial equations. By mastering exponent rules, tackling related mathematical challenges becomes much more manageable.