Problem 55 - Algebra A Combined Function

Question

Use the quotient rule and simplify each expression. $$ \frac{x^{3}}{x} $$

Step-by-Step Solution

Answer

The simplified expression is \( 2x \).

Step 1: Identify the Quotient

The given expression is \( \frac{x^3}{x} \). It is a quotient of two functions where the numerator is \( x^3 \) and the denominator is \( x \).

Step 2: Apply the Quotient Rule

To differentiate \( \frac{x^3}{x} \) using the quotient rule, we identify \( u = x^3 \) and \( v = x \). The quotient rule states \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \).

Step 3: Differentiate the Numerator and Denominator

Differentiate \( u = x^3 \) to get \( \frac{du}{dx} = 3x^2 \). Differentiate \( v = x \) to get \( \frac{dv}{dx} = 1 \).

Step 4: Substitute into the Quotient Rule

Substitute the derivatives into the quotient rule: \( \frac{x \cdot 3x^2 - x^3 \cdot 1}{x^2} \). This simplifies to \( \frac{3x^3 - x^3}{x^2} \).

Step 5: Simplify the Expression

Combine like terms in the numerator to get \( \frac{2x^3}{x^2} \). Simplify by dividing \( x^3 \) by \( x^2 \) to get \( 2x \).

Step 6: Result Verification

Verify that the simplification is correct by noting that each term that was manipulated has been reduced to its simplest form, confirming \( 2x \) is the simplified version.

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’s value changes as its input changes. In practical terms, it tells us how a function's output is affected by small changes in the input.

To differentiate functions that are expressed as a fraction or quotient, we often use the quotient rule. This is especially useful when the function can be written in the form \( \frac{u}{v} \), where both \( u \) and \( v \) are functions of \( x \).

• The quotient rule is given by: \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \).
• This formula allows us to find the derivative of the entire quotient by considering both the change in the numerator and the denominator.

Understanding differentiation is key to grasping how functions behave and to solving problems in calculus that involve rates of change.

Simplifying Expressions

Simplifying expressions is a crucial step in solving calculus problems. After applying differentiation techniques, especially with complex fractions or products, simplifying helps make the expression easier to interpret and use.

In our exercise, we initially have the expression \( \frac{x^3}{x} \). After differentiating using the quotient rule, the expression is transformed into \( \frac{3x^3 - x^3}{x^2} \).

• Combine like terms whenever possible, which in this case simplifies the numerator to \( 2x^3 \).
• Further simplify by canceling out common factors from the numerator and the denominator, giving us \( 2x \) as the simplest form.

Simplification not only makes the solution more transparent but also helps in verifying that the derived expression retains the same mathematical properties.

Functions

Functions are the building blocks of calculus. They define the relationship between two variables, typically \( x \) and \( y \). In the context of the exercise, the functions involved are \( u = x^3 \) and \( v = x \).

When tackling problems involving functions, it's helpful to understand:

• Each function’s domain, which tells you the set of possible input values.
• The range, indicating the set of possible output values.
• How the function behaves, such as increasing or decreasing trends.

Functions like \( u \) and \( v \) meet at "points of interest," where differentiation is often necessary to explore these behaviors' rates of change. Knowing how to handle and manipulate these functions is crucial for effective problem-solving in calculus scenarios like the exercise above.