Problem 18 - College Algebra and Calculus: An Applied Approach

Question

Match the function with the rule that you would use to find the derivative most efficiently. (a) Simple Power Rule (b) Constant Rule (c) General Power Rule (d) Quotient Rule $$ f(x)=\sqrt[3]{x^{2}} $$

Step-by-Step Solution

Answer

The most efficient rule to use for the function \( f(x)=\sqrt[3]{x^{2}} \) is the General Power Rule (c)

Step 1: Rewrite function in power form

Rewrite the function from root form to power form. The cubic root of \( x^{2} \) is equal to \( x^{2/3} \). So the function \( f(x)=\sqrt[3]{x^{2}} \) can be rewritten as \( f(x)=x^{2/3} \).

Step 2: Identify the rule for finding derivative

The function f(x) = \( x^{2/3} \) is a power function, where the base is x and the exponent is a real number. In this case, the exponent is 2/3. The rule to find the derivative of a power function is the General Power Rule, which states that the derivative of \( x^n \) is \( nx^{n-1} \).

Step 3: Match the function with the rule

Considering the form of the function \( f(x)=x^{2/3} \), it is clear that the most efficient rule to use when finding its derivative would be the General Power Rule

Key Concepts

Derivative

In calculus, a derivative represents the rate at which a function changes as its input changes. It can be thought of as the slope of the function at a particular point. Derivatives provide critical insight into various phenomena, from physics to finance.

To find a derivative, you study how changes in a function’s argument cause changes in the function’s value. This involves understanding the slope of the tangent line to the function at any given point. For example, if you have a function represented by the equation \( f(x) = x^{2/3} \), the derivative tells you how quickly \( f(x) \) is changing at any value of \( x \).

• It's a fundamental concept in calculus.
• Helps in calculating the slope of a curve.
• Essential in optimizing and analyzing changes.

The process of finding a derivative is commonly called differentiation. Each differentiation technique is based on specific rules for various types of functions.

General Power Rule

The General Power Rule is a core technique used in calculus to compute the derivative of functions that involve power expressions. Specifically, it applies to functions of the form \( f(x) = x^n \), where \( n \) can be any real number.

The mathematical expression of the General Power Rule is straightforward: \( \frac{d}{dx}[x^n] = nx^{n-1} \). This means you multiply the exponent by the coefficient of \( x \) and then decrease the exponent by one. For example, the derivative of \( x^{2/3} \) becomes \( \frac{2}{3}x^{-1/3} \).

• Works for positive, negative, zero, and fractional exponents.
• Simplifies the derivative process for power functions.
• Provides a quick method to find derivatives without complex steps.

Understanding the General Power Rule is essential for efficiently calculating the derivatives of various algebraic functions.

Power Function

A power function is a type of mathematical function that can be expressed in the form \( f(x) = x^n \), where \( n \) is a real number. These functions are of significant interest because they frequently appear in both mathematical theory and practical applications.

Power functions are characterized by their

• Simplicity and straightforward structure
• Ability to represent a wide range of behaviors depending on the exponent \( n \)
• Flexibility in modeling polynomial relationships and growth patterns

In the example \( f(x) = x^{2/3} \), the function is a power function because \( x \) is raised to a fractional power. Understanding these functions is crucial in calculus since they possess properties that simplify when differentiation rules, such as the General Power Rule, are applied.
Power functions help form the building blocks for more complex functions and are integral to problem-solving in calculus.