Problem 107 - Calculus Volume 1

Question

For the following exercises, find \(f^{\prime}(x)\) for each function. $$f(x)=5 x^{3}-x+1$$

Step-by-Step Solution

Answer

\( f'(x) = 15x^2 - 1 \)

Step 1: Understand the Problem

We need to find the derivative of the function \( f(x) = 5x^3 - x + 1 \). This means we want to calculate \( f'(x) \).

Step 2: Apply Derivative Rules

To find the derivative, apply the power rule: if \( f(x) = ax^n \), then \( f'(x) = anx^{n-1} \). Apply this rule to each term in the function \( f(x) = 5x^3 - x + 1 \).

Step 3: Differentiate Each Term

Differentiate the first term: \( \frac{d}{dx}(5x^3) = 15x^2 \). Differentiate the second term: \( \frac{d}{dx}(-x) = -1 \). The derivative of the constant \(1\) is \(0\).

Step 4: Combine the Derivatives

Combine the derivatives of each term to get \( f'(x) = 15x^2 - 1 \).

Key Concepts

Understanding the Power Rule

The power rule is one of the simplest and most commonly used techniques in calculus for finding the derivative of polynomial functions. It states that if you have a function of the form \( f(x) = ax^n \), then its derivative is \( f'(x) = anx^{n-1} \).

This rule helps us quickly determine how the function changes at any point \( x \). It is particularly handy because it breaks down complex polynomial expressions into manageable pieces.

• Constant Multiplication: The coefficient \( a \) is multiplied by the exponent \( n \).
• Exponent Reduction: The exponent \( n \) is decreased by one.

By applying the power rule, we can differentiate polynomial functions like \( 5x^3 - x + 1 \) with ease. Simply apply the rule to each term and sum the results.

The Process of Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative tells us the rate at which the function's output changes as its input changes — in other words, its slope or gradient.

When differentiating a polynomial function like our example \( f(x) = 5x^3 - x + 1 \), we handle each term separately:

• The derivative of \( 5x^3 \) is \( 15x^2 \) by the power rule.
• The derivative of \(-x\) is \(-1\), because the power of \( x \) is 1, which decrements to zero, leaving the coefficient.
• The derivative of a constant, like \(1\), is always \(0\) because constants do not change.

After finding the derivative of each term individually, you simply add them together to get the overall derivative \( f'(x) = 15x^2 - 1 \). Differentiation thus enables us to model dynamic systems and understand their behavior.

An Introduction to Calculus Concepts

Calculus is the branch of mathematics that studies continuous change. It provides us with tools to quantify how things evolve over time or space. Within calculus, two main operations stand out: differentiation and integration.

Differentiation, which you've now seen, focuses on rates of change — how fast or slow something occurs. It provides a mathematical method to calculate tangents or instantaneous rates.

• Understanding Change: Differentiation breaks down how a function grows or diminishes at any point.
• Analyzing Curves: Through differentiation, calculus reveals the behavior of functions, such as increasing or decreasing trends and concavity. Integration, the other fundamental calculus operation, deals with areas under curves or accumulating quantities.

Together, these concepts form the backbone of calculus, enabling us to solve real-world problems involving motion, growth, and more. They are essential in fields ranging from physics and engineering to economics and biology.