Problem 69 - Calculus and its applications

Question

Differentiate each function. \(f(x)=\frac{(x-1)\left(x^{2}+x+1\right)}{x^{4}-3 x^{3}-5}\)

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \frac{-x^6 + 4x^3 - 24x^2}{(x^4 - 3x^3 - 5)^2} \).

Step 1: Identify the derivative rule to use

The given function is a fraction or rational function of polynomials. To differentiate it, we'll use the quotient rule which states:\[ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \]where \( u \) is the numerator \( (x-1)(x^2+x+1) \) and \( v \) is the denominator \( x^4-3x^3-5 \).

Step 2: Differentiate the Numerator

First, expand the numerator function \( u=(x-1)(x^2+x+1) \) into a polynomial:\[ u = x^3 + x^2 + x - x^2 - x - 1 = x^3 - 1 \]Now, find the derivative of \( u \):\[ u' = \frac{d}{dx}(x^3 - 1) = 3x^2 \].

Step 3: Differentiate the Denominator

Differentiate the denominator function \( v = x^4 - 3x^3 - 5 \):\[ v' = \frac{d}{dx}(x^4 - 3x^3 - 5) = 4x^3 - 9x^2 \].

Step 4: Apply the Quotient Rule

Substitute \( u \), \( u' \), \( v \), and \( v' \) into the quotient rule formula:\[ f'(x) = \frac{(3x^2)(x^4 - 3x^3 - 5) - (x^3 - 1)(4x^3 - 9x^2)}{(x^4 - 3x^3 - 5)^2} \].

Step 5: Simplify the Derivative Expression

Expand and simplify the expression. Calculate the terms:1. \( 3x^2(x^4 - 3x^3 - 5) = 3x^6 - 9x^5 - 15x^2 \)2. \( (x^3 - 1)(4x^3 - 9x^2) = 4x^6 - 9x^5 - 4x^3 + 9x^2 \)Now combine:\[ f'(x) = \frac{3x^6 - 9x^5 - 15x^2 - 4x^6 + 9x^5 + 4x^3 - 9x^2}{(x^4 - 3x^3 - 5)^2} \]Simplify the expression:\[ f'(x) = \frac{-x^6 + 4x^3 - 24x^2}{(x^4 - 3x^3 - 5)^2} \].

Key Concepts

Derivative of Rational Functions

Rational functions are quotients where both the numerator and the denominator are polynomials. The process of finding their derivatives requires special rules beyond basic differentiation. The quotient rule is specifically designed to deal with such situations. It simplifies the process by providing a structured formula:

• For a given function \( f(x) = \frac{u}{v} \), the derivative \( f'(x) \) is calculated using the formula \( f'(x) = \frac{u'v - uv'}{v^2} \).
• Here, \( u \) and \( v \) represent the numerator and the denominator, respectively.

To apply the quotient rule, follow these steps:

• Differentiating the numerator \( u \) to get \( u' \).
• Differentiating the denominator \( v \) to get \( v' \).
• Then substitute \( u \), \( u' \), \( v \), and \( v' \) into the quotient rule formula.

This technique is vital to properly manage the derivatives of rational functions without mistakenly multiplying or dividing components incorrectly.

Polynomial Differentiation

Polynomials are expressions consisting of variables raised to whole number powers and their constant coefficients. The differentiation of polynomials is foundational, involving rules such as the power rule. The process is fairly straightforward:

• For a term \( ax^n \), its derivative is found using the power rule: \( \frac{d}{dx}(ax^n) = nax^{n-1} \).
• Each term in the polynomial is differentiated individually, and then they're combined together again.

When we differentiate the polynomial numerator \( (x-1)(x^2+x+1) \) as seen in our example:

• First, it is expanded to a single polynomial \( x^3 - 1 \).
• Then differentiated term by term, here yielding \( 3x^2 \) for the derivative.

Similarly, the denominator \( x^4-3x^3-5 \) is handled by identifying each term and finding the derivatives to get \( 4x^3 - 9x^2 \). Understanding and applying polynomial differentiation is crucial for efficiently dealing with both the numerator and the denominator of rational functions.

Simplifying Derivatives

After applying the quotient rule, simplifying the resulting expression can often be the most challenging yet crucial step. Simplification involves systematically reducing the expression to its simplest form:

• Begin by carefully expanding products using the distributive property. In our example: \( 3x^2(x^4 - 3x^3 - 5) \) becomes \( 3x^6 - 9x^5 - 15x^2 \).
• Follow similarly for the other expansion \((x^3 - 1)(4x^3 - 9x^2)\).
• Combine like terms from these expansions to finalize the numerator.

In our transformed exercise, this leads to combining terms through addition and subtraction, ultimately simplifying the numerator to \(-x^6 + 4x^3 - 24x^2\). Simplification reduces computational complexity and makes the expression more comprehensible. It ensures accuracy and clarity, especially when dealing directly with higher-degree polynomials.