Problem 1 - Brief Applied Calculus

Question

Find the derivative of each function. $$ f(x)=x^{4} $$

Step-by-Step Solution

Answer

The derivative of \( f(x) = x^4 \) is \( f'(x) = 4x^3 \).

Step 1: Understand the Power Rule

To find the derivative of a function like \( f(x) = x^n \), we use the Power Rule of differentiation. The Power Rule states that if \( f(x) = x^n \), then the derivative \( f'(x) \) is given by \( f'(x) = nx^{n-1} \). In this exercise, you will apply this rule to find the derivative of the function \( f(x) = x^4 \).

Step 2: Identify the Power and Coefficient

In the function \( f(x) = x^4 \), identify the exponent \( n \). Here, \( n = 4 \). There is an implicit coefficient of \( 1 \) in front of \( x^4 \), which is essential for using the Power Rule.

Step 3: Apply the Power Rule

Using the Power Rule, multiply the coefficient 1 by the exponent 4, and reduce the exponent by 1. This yields: \( f'(x) = 4 \cdot x^{4-1} \).

Step 4: Simplify the Derivative

Simplify the expression obtained from applying the Power Rule. The expression \( 4 \cdot x^{4-1} \) simplifies to \( 4x^3 \). Therefore, the derivative of the function \( f(x) = x^4 \) is \( f'(x) = 4x^3 \).

Key Concepts

Power Rule

The Power Rule is a fundamental concept in calculus that greatly simplifies finding the derivative of functions raised to a power. It states: if you have a function in the form \( f(x) = x^n \), its derivative \( f'(x) \) is \( nx^{n-1} \). When applying the Power Rule:

• Identify the exponent \( n \).
• Multiply the entire function by the exponent.
• Subtract one from the original exponent to form the new power of \( x \).

This rule is particularly useful for polynomial functions, where each term can be individually differentiated. This makes it a cornerstone of calculus education, providing a simple method to tackle more complex differentiation problems. As a quick example: for \( f(x) = x^4 \), using the Power Rule gives us \( f'(x) = 4x^{3} \).

Differentiation

Differentiation is the process used in calculus to find the rate at which a function is changing at any given point. Essentially, it tells us how steep the graph of a function is at a particular location. This is applicable to many real-world scenarios, such as speed or growth rate calculations. When differentiating a function like \( f(x) = x^4 \), you apply rules such as the Power Rule to systematically find the derivative. This derivative, represented as \( f'(x) \) or sometimes \( \frac{df}{dx} \), describes the slope of the tangent line to the function at any point \( x \). In our simple example, differentiation tells us that for \( f(x) = x^4 \), the slope is \( 4x^3 \). This new function \( f'(x) = 4x^3 \) becomes crucial for analysis and optimization purposes.

Functions

Functions in mathematics describe relationships between sets of numbers or objects. They serve as a basic building block of calculus, often denoted as \( f(x) \). A function assigns each input \( x \) to exactly one output \( f(x) \).Understanding the nature of a function is essential for applying calculus principles such as differentiation. For a function \( f(x) = x^4 \), every value of \( x \) will produce a corresponding value of \( x \) raised to the fourth power. This offers a predictable pattern in the data, allowing us to apply rules like the Power Rule for derivatives.Knowing the behavior of a function also helps when graphing relationships or modeling real-world phenomena. Functions can be linear or nonlinear, simple or complex, and serve as a representation of natural occurrences or theoretical models.