Simplify the difference quotient (ƒ(x)-ƒ(a)) / (x-a) for the following functions.
ƒ(x) = x⁴

Verified step by step guidance

1. First, understand the difference quotient: it is a formula used to find the average rate of change of a function between two points, x and a. It is given by (ƒ(x) - ƒ(a)) / (x - a).
2. Substitute the given function ƒ(x) = x⁴ into the difference quotient formula. This gives us: ((x⁴) - (a⁴)) / (x - a).
3. Recognize that the expression x⁴ - a⁴ is a difference of powers, which can be factored using the formula for the difference of two squares: x⁴ - a⁴ = (x² - a²)(x² + a²).
4. Further factor x² - a² using the difference of squares formula: x² - a² = (x - a)(x + a). Therefore, x⁴ - a⁴ = (x - a)(x + a)(x² + a²).
5. Cancel the common factor (x - a) from the numerator and the denominator of the difference quotient, resulting in the simplified expression: (x + a)(x² + a²).

Key Concepts

Difference Quotient

The difference quotient is a formula used to calculate the average rate of change of a function over an interval. It is expressed as (ƒ(x) - ƒ(a)) / (x - a), where ƒ(x) is the function and a is a specific point. This concept is foundational in calculus as it leads to the definition of the derivative, which represents the instantaneous rate of change.

Polynomial Functions

Polynomial functions are mathematical expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. The function ƒ(x) = x⁴ is a polynomial of degree four, meaning its highest exponent is four. Understanding the properties of polynomial functions, such as their behavior and derivatives, is crucial for simplifying expressions like the difference quotient.

Limit and Derivative

In calculus, the limit is a fundamental concept that describes the behavior of a function as it approaches a certain point. The derivative, which is defined as the limit of the difference quotient as x approaches a, provides the instantaneous rate of change of the function at that point. Simplifying the difference quotient for ƒ(x) = x⁴ will ultimately lead to finding its derivative, illustrating the connection between these concepts.