Problem 13 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=x^{4}+x^{3}+x^{2}+x+1 $$

Step-by-Step Solution

Answer

The derivative \( D_x y \) is \( 4x^3 + 3x^2 + 2x + 1 \).

Step 1: Identify the Function

The function given is \( y = x^4 + x^3 + x^2 + x + 1 \). This is a polynomial function of degree 4.

Step 2: Apply the Power Rule

To find the derivative \( D_x y \), we apply the power rule, which states that the derivative of \( x^n \) is \( nx^{n-1} \).

Step 3: Differentiate Each Term

Differentiate each term of the polynomial separately: - The derivative of \( x^4 \) is \( 4x^3 \) using the power rule.- The derivative of \( x^3 \) is \( 3x^2 \).- The derivative of \( x^2 \) is \( 2x \).- The derivative of \( x \) is \( 1 \), since \( x^1 \) becomes \( 1x^0 = 1 \).- The derivative of a constant \( 1 \) is \( 0 \), since the derivative of any constant is zero.

Step 4: Write the Derivative

Now, combine all the differentiated terms to get the derivative of the function: \[ D_x y = 4x^3 + 3x^2 + 2x + 1 \]

Key Concepts

Understanding Polynomial Functions

A polynomial function is an expression made up of terms where each term is a constant multiplied by a variable raised to a non-negative integer power. For example, the polynomial function in our exercise is given by: \( y = x^4 + x^3 + x^2 + x + 1 \). In this function, each term features the variable \( x \) raised to a power ranging from 0 to 4. Polynomial functions are very common in mathematics, especially in calculus, because they are smooth and continuous. This means they stretch out in gentle curves over a graph without breaks or sharp turns. The degree of the polynomial, which is the highest power of the variable, determines the overall shape and behavior of its graph. For our function, the degree is 4, suggesting a relatively complex shape.

Mastering the Power Rule

The Power Rule is a quick and handy tool for differentiation in calculus. It states that if you have a function \( x^n \), its derivative is \( nx^{n-1} \). This means you multiply the original exponent \( n \) by the coefficient of the term and reduce the exponent by one. Let's apply this rule to a simple term, like \( x^3 \).

• First, multiply 3 (the exponent) by 1 (the coefficient of the term) to get 3.
• Then, decrease the exponent by 1, resulting in \( 3x^2 \).

Applying this power rule to each term of a polynomial function allows us to efficiently find its derivative. In our exercise, we used it systematically on each component term of the polynomial.

A Glimpse into Calculus

Calculus is a branch of mathematics focused on the concepts of change and motion. It provides tools to understand how functions increase, decrease, bend, or maintain a constant rate. Two primary operations in calculus are differentiation and integration. Differentiation, which we focus on here, deals with finding the derivative of a function, which indicates the rate of change or slope of the function at any given point. With these insights, you can predict how a variable relationship evolves over time. In our problem, differentiation helps us uncover how the function \( y = x^4 + x^3 + x^2 + x + 1 \) changes with small movements in \( x \).

The Art of Differentiation

Differentiation is the process of finding the derivative of a function. It's one of the core techniques of calculus that allows us to analyze the behavior of functions. The derivative tells us the rate at which a function is changing at any point along its curve. In our example, once each term in the polynomial \( y = x^4 + x^3 + x^2 + x + 1 \) is differentiated independently using the power rule, the derivatives are gathered to form \( D_x y = 4x^3 + 3x^2 + 2x + 1 \).

• Each differentiated term corresponds to the slope of the curve at that term's influence.
• The sum of these differentiated terms gives the overall instantaneous rate of change of the function.

Differentiation is thus a tool for dissecting functions to better understand their dynamic nature.