Problem 13 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(D_{x} y\) using the rules of this section. $$ y=x^{4}+x^{3}+x^{2}+x+1 $$

Step-by-Step Solution

Answer

\(D_{x} y = 4x^3 + 3x^2 + 2x + 1\).

Step 1: Given function

We need to find the derivative of the function with respect to \(x\). The function is \(y = x^4 + x^3 + x^2 + x + 1\).

Step 2: Apply Power Rule

The power rule for derivatives states that if \(y = x^n\), then \(\frac{d}{dx}x^n = nx^{n-1}\). We will apply this rule to each term of the polynomial function.

Step 3: Differentiate each term

1. Differentiate \(x^4\): \(\frac{d}{dx}(x^4) = 4x^{3}\).2. Differentiate \(x^3\): \(\frac{d}{dx}(x^3) = 3x^{2}\).3. Differentiate \(x^2\): \(\frac{d}{dx}(x^2) = 2x\).4. Differentiate \(x\): \(\frac{d}{dx}(x) = 1\).5. Differentiate the constant 1, which gives \(\frac{d}{dx}(1) = 0\).

Step 4: Summing derivatives

Combine all the derived terms to get the derivative of the entire function. This gives us \(D_{x} y = 4x^3 + 3x^2 + 2x + 1\).

Key Concepts

Power Rule

The power rule is one of the fundamental techniques in calculus to find the derivative of a function. It helps simplify the process, especially for functions where the variable is raised to a power, like polynomial functions.
It states that if you have a function in the form of \(y = x^n\), to find its derivative, \ \(\frac{d}{dx}x^n = nx^{n-1}\ \).
This means you take the exponent \(n\), bring it in front of the term to multiply, and then subtract one from the exponent.

Power rule applies to each term separately in a multi-term function, making it easy to manage and compute the derivative for complex polynomial functions. It is a foundational rule that is frequently used in calculus.

Polynomial Functions

Polynomial functions are expressions consisting of terms formed by variables raised to a positive integer exponent, multiplied by coefficients. They are essentially sums of one or more monomials.

• The general form of a polynomial function is given by \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where \(a_n\) are coefficients and \(x\) is the variable.
• In the original exercise, the function \(y = x^4 + x^3 + x^2 + x + 1\) is a polynomial.

Each term in a polynomial can be differentiated independently using rules like the power rule.
Polynomial functions are among the simplest to differentiate because each term is straightforward to compute using basic differentiation techniques.

Differentiation Techniques

Differentiation techniques are a set of rules and methods applied to find the derivative of a function. At the core is understanding how these techniques work and when to apply them. Here are some key methods:

• Power Rule: This is used when each term of the function is a power of \(x\). We differentiate each term separately.
• Constant Rule: For a constant term like '1' in our equation, the derivative is zero because constants do not change when \(x\) changes.

In the case of polynomial functions, these techniques allow you to systematically figure out the derivative by applying rules to each term. Differentiation serves as the backbone of calculus and is used for finding rates of change, slopes of tangent lines, and more.
Understanding which technique to employ based on the function's form is crucial in simplifying and accurately deriving the required results.