Problem 32 - College Algebra and Calculus: An Applied Approach

Question

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ f(x)=\frac{x^{3}+3 x+2}{x^{2}-1} $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = (x^3 + 3x + 2)/(x^2 -1)\) is \(f'(x) = (x^4 -9x^2 - 4x -3) / (x^4 -2x^2 + 1)\)

Step 1: Identify u and v

First, the function needs to be rewritten in the form of \(u(x)/v(x)\). Identify the numerator \(u(x) = x^3 + 3x + 2\) and the denominator \(v(x) = x^2 -1\).

Step 2: Calculate derivate of u(x) and v(x)

We know that the derivative of \(u(x)\) is \(u'(x)\) and the derivative of \(v(x)\) is \(v'(x)\). Using the power rule, \(u'(x) = 3x^2 + 3\) and \(v'(x) = 2x\).

Step 3: Use the quotient rule

Using the quotient rule for differentiation, we get \(f'(x) = (u'(x)v(x) - u(x)v'(x)) / v^2(x)\). Where \(u'(x) = 3x^2 + 3\), \(v(x) = x^2 -1\), \(u(x) = x^3 + 3x + 2\) and \(v'(x) = 2x\). Substituting these values we get, \(f'(x) = ((3x^2 + 3)(x^2 -1) - (x^3+3x+2)(2x)) / (x^2 -1)^2\). This simplifies to \(f'(x) = (3x^4 -3x^2 +3x^2 -3 -2x^4 -6x^2 -4x) / (x^4 -2x^2 + 1)\) which further simplifies to \(f'(x) = (x^4 -9x^2 - 4x -3) / (x^4 -2x^2 + 1)\)

Key Concepts

Derivative Calculations

Derivative calculations are fundamental in calculus, as they measure how a function changes as its input changes. Simply put, the derivative tells us the rate at which one quantity is changing with respect to another.

To find a derivative, various rules can be applied depending on the form of the function. For instance, when given a ratio of two functions, like in the exercise with the function \( f(x)=\frac{x^{3}+3x+2}{x^{2}-1} \), the quotient rule for differentiation is particularly useful. This rule provides a formula to derive the function by considering derivatives of both the numerator and the denominator functions separately, and then combining the results in a specific manner.

Power Rule

The power rule is a quick and efficient tool for differentiating polynomial functions where the variable x is raised to a power. The rule states that if \( f(x) = x^n \), where n is a real number, then the derivative \( f'(x) = nx^{n-1} \).

In the given exercise, this rule was utilized to differentiate the polynomial functions in the numerator \( u(x) = x^3 + 3x + 2 \) and the denominator \( v(x) = x^2 - 1 \). Applying the power rule, we find \( u'(x) = 3x^2 + 3 \) and \( v'(x) = 2x \). The simplicity of the power rule makes calculations faster and more straightforward, especially for polynomial functions.

Differentiating Polynomial Functions

Differentiating polynomial functions involves finding the derivative of functions expressed as sums of powers of the variable. These functions are usually in the form \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \), where each a_n represents a coefficient.

With each term being a power function, the power rule can be applied to each term individually. For example, in the numerator \( u(x) = x^3 + 3x + 2 \) of our original function \( f(x) \), we differentiate each term to find \( u'(x) \), resulting in \( 3x^2 + 3 \). Polynomial functions' derivatives are also polynomial, which maintains the simplicity and elegance of the expressions through the differentiation process.

Calculus for Algebraic Functions

Algebraic functions, like the one in our exercise, include polynomials, rational functions, and roots of polynomials. Calculus plays a crucial role in understanding the behavior of these functions through differentiation and integration.

The given function \( f(x) = \frac{x^{3}+3x+2}{x^{2}-1} \) is a rational function, which is the ratio of two polynomials. To differentiate such functions, we need a combination of rules, predominantly the quotient rule, as seen in the solution, and potentially the power rule, as used to differentiate the numerator and denominator. These rules and their proper application in calculus allow us to analyze and interpret algebraic functions in terms of their slopes, rates of change, and overall behavior.