Problem 32 - 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 \(f(x) = \frac{x^{3} + 3x + 2}{x^{2} - 1}\) is \(f'(x) = \frac{5x^{2} + 3}{(x^{2} - 1)^{2}}\)

Step 1: Identify the Numerator and Denominator

Rewrite the function and identify the numerator and denominator. Here, the numerator \(u(x) = x^{3} + 3x + 2\) and the denominator \(v(x) = x^{2} - 1\).

Step 2: Differentiate Numerator and Denominator

Find the derivative of the numerator and denominator separately. The derivative of the numerator \(u'(x) = 3x^{2} + 3\) and the derivative of the denominator \(v'(x) = 2x\).

Step 3: Apply the Quotient Rule

Apply the quotient rule for derivatives, which states that the derivative of a quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator all over the denominator squared. Substituting into the formula gives us \(f'(x) = \frac{(x^{2} - 1)(3x^{2} + 3) - (x^{3} + 3x + 2)(2x)}{(x^{2} - 1)^{2}}\).

Step 4: Simplify

Simplify to get the final result. Expanding the right side and simplifying terms, we get \(f'(x) = \frac{5x^{2} + 3}{(x^{2} - 1)^{2}}\).

Key Concepts

Derivative Calculus

Derivative calculus lies at the heart of mathematics, offering insight into the rate at which quantities change. It is fundamental to the study of calculus and underpins much of the advanced mathematics used in engineering, science, and economics. A derivative measures how a function's output value changes as its input value changes.

In essence, when we calculate the derivative of a function, such as \( f(x) \), we are determining the function's rate of change or slope at every point along its curve. This is often visualized as the slope of the tangent line to the function's graph at a specific point. The process of finding a derivative is called differentiation.

For the given function \( f(x) = \frac{x^{3}+3x+2}{x^{2}-1} \), the derivative conveys how this ratio of polynomials changes with respect to \( x \). Students should realize that the derivative reflects the instantaneous rate of change, and they can calculate it using various rules and techniques of differentiation.

Differentiation Techniques

When confronted with the challenge of finding the derivative of a complex function, mathematicians employ a variety of differentiation techniques. Each technique applies to specific types of functions or combinations thereof.

Some common techniques include:

• The Power Rule: Easily find the derivative of monomials, like \( x^n \).
• The Product Rule: Use when differentiating the product of two functions.
• The Chain Rule: Apply to composite functions—functions of functions.
• The Quotient Rule: As seen in the exercise, apply when differentiating a function that is the quotient of two other functions.

The quotient rule, used in our example, is particularly handy. It's expressed as \( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v u' - u v'}{v^2} \), where \( u \), and \( v \), represent the numerator and denominator of our function, respectively, and \( u' \) and \( v' \) their derivatives. By breaking down complex expressions into simpler parts and applying the quotient rule, differentiation becomes a more approachable task.

Simplifying Derivatives

Simplifying derivatives is a pivotal step to derive the most compact and usable form of a derivative. After using differentiation rules, we often end with a complex expression that can be simplified by combining like terms, factoring, or canceling terms.

Consider our given function's derivative: \( f'(x) = \frac{(x^{2} - 1)(3x^{2} + 3) - (x^{3} + 3x + 2)(2x)}{(x^{2} - 1)^{2}} \). Simplifying such an expression requires patience and careful algebraic manipulation. For instance, expanding and combining like terms might lead to cancellations that pare down the derivative to a more manageable form.

In the final step of our problem, we achieve the simplified form \( f'(x) = \frac{5x^{2} + 3}{(x^{2} - 1)^{2}} \). This step is crucial because it makes the result clearer and often more practical, especially when evaluating the derivative at particular points. Simplification is not just about making the expression shorter; it's about enhancing the readability and comprehensibility of the derivative for further applications.