Problem 104 - Calculus and its applications

Question

Find dy/dx. Each function can be differentiated using the rules developed in this section, but some algebra may be required beforehand. $$ y=(x-1)(x+1) $$

Step-by-Step Solution

Answer

The derivative \( \frac{dy}{dx} = 2x \).

Step 1: Expand the Expression

Before differentiating, we'll expand the expression for \( y \). We have \( y = (x-1)(x+1) \). Using the distributive property (FOIL method), we get: \( y = x^2 + x - x - 1 = x^2 - 1 \).

Step 2: Differentiate the Expanded Function

Now that we have \( y = x^2 - 1 \), we can differentiate this function with respect to \( x \). The derivative of \( x^2 \) is \( 2x \), and the derivative of \(-1\) is \( 0 \). Therefore, \( \frac{dy}{dx} = 2x \).

Key Concepts

Algebraic Expansion

Algebraic expansion is a foundational concept in mathematics. It involves the process of multiplying and distributing terms to simplify expressions, often making them easier to work with in various operations like differentiation and integration. In the context of the given exercise, algebraic expansion is crucial. Instead of differentiating the product directly, we expand the expression

• The term "algebraic expansion" specifically refers to replacing a product of two or more expressions with a single expanded expression.
• This simplification helps us use basic differentiation rules easily.
• Taking the original egin{equation}y = (x-1)(x+1), ext{ we expand } it.} { This involves distributing each term from (x-1) and (x+1): x \cdot x, x \cdot 1, -1 \cdot x, -1 \cdot 1.

With this, the expression becomes \( x^2 - 1 \).In mathematical analysis, expanding expressions this way is always helpful before taking derivatives when complex terms are involved. It simplifies the subsequent steps significantly.
This algebraic technique is particularly useful when dealing with polynomials and functions involving multiple terms.

Distributive Property

The distributive property is a basic but powerful tool in mathematics that allows us to simplify and rearrange expressions. The property states that multiplying a number by a sum is the same as doing each multiplication separately and then adding the results. In symbolic form, this is shown as \( a(b + c) = ab + ac \). This principle applies to polynomials as well.

• The distributive property was used in our example when expanding \( (x-1)(x+1) \).
• We applied the property to combine terms resulting in the polynomial \( x^2 - 1 \).
• Applying the distributive property simplifies expressions, making them more manageable for further operations like differentiation.

By distributing each term, complex expressions become simplified into a form that is easier to differentiate. This concept lies at the heart of algebraic manipulation, underlying many processes in calculus and beyond.

FOIL Method

The FOIL method is a special application of the distributive property, primarily used for expanding binomials. FOIL stands for First, Outer, Inner, Last, representing the order in which you multiply terms in two binomials \((a+b)(c+d)\). In the given problem, the FOIL method was used to expand \((x-1)(x+1)\). Here's how it works:

• First: Multiply the first terms in each binomial, resulting in \( x \cdot x = x^2 \).
• Outer: Multiply the outer terms, yielding \( x \cdot 1 = x \).
• Inner: Multiply the inner terms, which gives \(-1 \cdot x = -x \).
• Last: Multiply the last terms in each binomial, resulting in \(-1 \cdot 1 = -1 \).

When these products are summed, we simplify to get \( x^2 - x + x - 1 = x^2 - 1 \).
The FOIL method helps ensure no terms are omitted during expansion, making it easier for students to tackle polynomial multiplication. By converting an expression into a simplified polynomial, differentiation then becomes straightforward, maximizing efficiency.