Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=x^{2}+2 x $$

Step-by-Step Solution

Answer

The derivative \( D_x y \) is \( 2x + 2 \).

Step 1: Identify the Function

We need to differentiate the function \( y = x^2 + 2x \). Here, it's a polynomial function.

Step 2: Apply the Power Rule

To differentiate \( x^n \), use \( \frac{d}{dx}x^n = nx^{n-1} \). For \( x^2 \), \( n = 2 \), so \( \frac{d}{dx}x^2 = 2x^{2-1} = 2x \).

Step 3: Differentiate the Linear Term

By applying the constant multiplication rule, the differentiation of \( 2x \) is \( 2 \times 1 = 2 \).

Step 4: Combine the Derivatives

Add the derivatives from steps 2 and 3: \( D_x y = 2x + 2 \).

Key Concepts

Polynomial Functions

Polynomial functions are one of the simplest types of functions in calculus and algebra. They are expressions consisting of variables and coefficients, structured in terms of powers. For example, the function we are working with, \( y = x^2 + 2x \), is a second-degree polynomial because the highest power of \( x \) in the expression is 2. This type of polynomial is known as a quadratic polynomial.
Understanding polynomial functions is crucial because they form the basis for more complex functions. Here are some characteristics of polynomial functions:

• They have terms in the form of \( ax^n \) where \( a \) is a coefficient and \( n \) is a non-negative integer.
• Polynomial functions are smooth and continuous; they do not have breaks, holes, or sharp turns.
• The degree of the polynomial gives us important information about the function's shape and number of roots.

This foundational knowledge is essential when performing operations such as differentiation.

Power Rule

The Power Rule is a fundamental tool for differentiating polynomial functions. It allows us to find the derivative of any term of the form \( x^n \) with ease. The rule states that if you have a term \( x^n \), where \( n \) is any real number, the derivative is \( n \times x^{n-1} \). For example, for \( x^2 \), applying the Power Rule results in \( 2x^{2-1} = 2x \).
This rule is particularly helpful because:

• It simplifies finding derivatives of terms with different powers without having to perform complex algebraic manipulations.
• The Power Rule can be effortlessly applied to each term individually in a polynomial expression, making the differentiation process straightforward and quick.

Mastery of the Power Rule is essential for efficiently tackling calculus problems involving polynomial derivatives.

Differentiation

Differentiation is the process of finding the derivative of a function. The derivative represents the rate of change of a function concerning its variable. In simple terms, it tells us how the function \( y \) changes as \( x \) changes. For the function \( y = x^2 + 2x \), finding the derivative \( D_x y \) involves determining how \( y \) changes at any value of \( x \).
Here’s why differentiation is important in calculus:

• It helps in understanding the behavior of functions, including their growth, decline, and stationary points.
• Differentiation is applied in numerous fields like physics for motion analysis, economics for cost functions, and biology for growth patterns.
• It provides the foundation for solving differential equations and understanding integration, another core calculus concept.

This pivotal concept enhances one's ability to predict and analyze real-world phenomena.

Calculus

Calculus is a vast field of mathematics focused on limits, functions, derivatives, integrals, and infinite series. It is divided mainly into two branches: differential calculus and integral calculus. In our exercise, we are concentrating on differential calculus, which involves finding the rate of change of a function.
The importance of calculus cannot be overstated. Here are some key aspects:

• Calculus allows us to model and solve problems scientifically and economically, where change is involved.
• It provides tools for optimizing and predicting future trends, making it indispensable in engineering, physics, probability, and many other scientific disciplines.
• Calculus equips learners with analytical techniques to handle complex problems by breaking them down into simpler parts.

Whether you are looking at the trajectory of a rocket or minimizing costs in a business, calculus offers the insights needed to make informed decisions.