Problem 2 - Calculus

Question

Find the derivatives of the functions in \(1-26\). $$ \left(x^{2}+1\right)\left(x^{2}-1\right) $$

Step-by-Step Solution

Answer

The derivative is \( 4x^3 \).

Step 1: Identify the Function Type

The given function is \( (x^2+1)(x^2-1) \), which is a product of two polynomials. We must use the product rule to differentiate.

Step 2: Recall the Product Rule

The product rule states that if \( u(x) \) and \( v(x) \) are functions of \( x \), then the derivative of their product is \( (uv)' = u'v + uv' \).

Step 3: Set Up Functions and Derivatives

Let \( u(x) = x^2 + 1 \) and \( v(x) = x^2 - 1 \). We need to find \( u'(x) \) and \( v'(x) \).- \( u'(x) = \frac{d}{dx}(x^2 + 1) = 2x \).- \( v'(x) = \frac{d}{dx}(x^2 - 1) = 2x \).

Step 4: Apply Product Rule

Substitute \( u, u', v, \) and \( v' \) into the product rule formula:\[ \frac{d}{dx}((x^2+1)(x^2-1)) = (x^2+1)(2x) + (x^2-1)(2x) \].

Step 5: Distribute and Simplify

Distribute \( 2x \) into both terms:- First term: \( (x^2+1)(2x) = 2x^3 + 2x \).- Second term: \( (x^2-1)(2x) = 2x^3 - 2x \).Combine the results:\[ 2x^3 + 2x + 2x^3 - 2x \].

Step 6: Combine Like Terms

Combine the like terms:- \( 2x^3 + 2x^3 = 4x^3 \).- \( 2x - 2x = 0 \).The simplified expression is \( 4x^3 \).

Key Concepts

Product Rule

The product rule is a fundamental principle in calculus used when differentiating the product of two functions. If you have two functions, say \( u(x) \) and \( v(x) \), their derivative can be found using the product rule formula:

• \( (uv)' = u'v + uv' \)

This means you differentiate the first function \( u(x) \) to get \( u'(x) \) and multiply it with the second function \( v(x) \), and vice versa. Then, you sum the two results.
This rule is essential when working with products because it accounts for both functions' influences on the overall product's rate of change. In our example, we applied the product rule to differentiate \((x^2 + 1)(x^2 - 1)\).
By setting \( u(x) = x^2 + 1 \) and \( v(x) = x^2 - 1 \), the problem becomes manageable, leading us to the derivative \( 4x^3 \), showcasing how the rule applies practically.

Polynomials

Polynomials are expressions involving variables raised to whole number powers, typically written in the form \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \). In this problem, each of the functions \( x^2+1 \) and \( x^2-1 \) is a polynomial.
These particular polynomials are quadratic, meaning their highest exponent is 2. Polynomials are widely used in calculus because they are relatively straightforward to differentiate and integrate.
While a single polynomial might be easy to handle, product rules and other operations are necessary when they are combined, as with the product of \( (x^2+1) \) and \( (x^2-1) \).
This combination of polynomials forms a new expression that's still structurally a polynomial, but it requires techniques like the product rule for proper differentiation.

Differentiation

Differentiation is the process of finding a derivative, which represents how a function changes as its input changes. It's a core concept in calculus.
In our problem, differentiation helps us find the rate at which the function \( (x^2+1)(x^2-1) \) changes with respect to \( x \).
By breaking down the function into its components using the product rule, we discovered that the derivative of the product results in a much simpler expression: \( 4x^3 \).
This simplification demonstrates the power of differentiation, as it allows you to transform complex function interactions into more manageable forms, leading to easier analysis in applications like physics, engineering, and economics.

Functions of x

A function of \( x \) is any expression where \( x \) is treated as the variable. Functions of \( x \) can take many forms, such as polynomial, exponential, logarithmic, and more.
In the current context, both \( (x^2+1) \) and \( (x^2-1) \) are functions of \( x \).

• They describe a relationship where each input \( x \) corresponds to a specific output.
• The composite function formed by their product \( (x^2+1)(x^2-1) \) further illustrates the complexity that can arise when combining multiple simple functions.

Understanding functions of \( x \) is crucial in calculus as it allows us to explore and analyze changes and interactions between various mathematical expressions.
These are foundational skills for any calculus student aiming to tackle real-world problems.