Problem 22 - College Algebra and Calculus: An Applied Approach

Question

Match the function with the rule that you would use to find the derivative most efficiently. (a) Simple Power Rule (b) Constant Rule (c) General Power Rule (d) Quotient Rule $$ f(x)=\frac{5}{x^{2}+1} $$

Step-by-Step Solution

Answer

The most efficient rule to find the derivative of the function \(f(x)=\frac{5}{x^{2}+1}\) is the Quotient Rule (d).

Step 1: Identify the structure of the function

The function \(f(x)=\frac{5}{x^{2}+1}\) is a rational function where the numerator is a constant and the denominator is a polynomial function.

Step 2: Match with the most suitable rule

Given the structure of the function, the Quotient rule is the most appropriate to use. The Quotient rule is used when finding the derivative of a fraction where both the numerator and the denominator are functions.

Key Concepts

Quotient Rule

The quotient rule is essential for calculating derivatives of fractions where both the numerator and the denominator are functions. It is particularly useful when the function can be expressed in the form \( \frac{g(x)}{h(x)} \). This rule is applied when you can't simplify the expression easily to use simpler derivative rules.

The formula for the quotient rule is:

• \( \left( \frac{g(x)}{h(x)} \right)' = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} \)

Here's how the formula works:

• \( g(x) \) is the numerator function and \( h(x) \) is the denominator function.
• \( g'(x) \) and \( h'(x) \) represent the derivatives of \( g(x) \) and \( h(x) \), respectively.

When working through problems, make sure you carefully compute each derivative separately and then substitute them back into the formula. This prevents mistakes and ensures clarity. A common error is forgetting to square the denominator, so keep an eye out for that step!

Rational Functions

Fun fact, a rational function is a fraction where both the numerator and the denominator are polynomials. It means the function is of the form \( \frac{p(x)}{q(x)} \). Rational functions consist of values that are well-defined except where the denominator is zero, as division by zero is undefined.

These functions can model many real-world situations, such as rates and proportions. In calculus, we often find their derivatives to analyze and understand the behavior of the function.

• The domain of rational functions excludes values that make the denominator zero.
• If a rational function's numerator degree is less than the denominator's, the horizontal asymptote is at 0.
• For the derivative, leveraging the quotient rule is most effective unless further simplification is possible.

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole number powers and their corresponding coefficients. They are some of the simplest kinds of functions to differentiate. Examples include linear functions, quadratics, and more complex expressions with higher degrees.

Key features of polynomial functions:

• The degree of a polynomial function is the highest power of the variable present.
• These functions are continuous and smooth, meaning they have no breaks or sharp corners.
• They are defined for all real numbers, unlike rational functions that may have restrictions.

To differentiate polynomial functions, use the power rule:

• For each term \( ax^n \), the derivative is \( nax^{n-1} \).

This rule simplifies the process of finding derivatives of polynomial functions, making them quick to solve when isolated, or part of more complex expressions like rational functions.