Problem 56 - Brief Applied Calculus

Question

The following problems extend and augment the material presented in the text. Find a general formula for \(\frac{d^{n}}{d x^{n}} x^{-1}\).

Step-by-Step Solution

Answer

\( \frac{d^n}{dx^n} x^{-1} = (-1)^n (n-1)! x^{-(n+1)} \).

Step 1: Understand the Base Condition

To find the general formula for \( \frac{d^n}{dx^n} x^{-1} \), we need to understand the base condition, which is the first derivative. The first derivative is \( \frac{d}{dx} x^{-1} = -x^{-2} \).

Step 2: Identify the Pattern

Let's look for a pattern by calculating the next few derivatives manually. The second derivative is \( \frac{d^2}{dx^2} x^{-1} = \frac{d}{dx}(-x^{-2}) = 2x^{-3} \). Continuing this, the third derivative \( \frac{d^3}{dx^3} x^{-1} = \frac{d}{dx}(2x^{-3}) = -6x^{-4} \).

Step 3: Generalize the Pattern

Notice the coefficients in the derivatives: -1, 2, -6. This sequence corresponds to the factorial sequence with alternating signs: \((-1)^{n-1}(n-1)!\). The general derivative becomes \((-1)^n (n-1)! x^{-(n+1)}\).

Step 4: Formulate the General Formula

From the pattern, we can conclude that the nth derivative is \( \frac{d^n}{dx^n} x^{-1} = (-1)^n (n-1)! x^{-(n+1)} \). This accounts for the alternating signs and the factorial growth of the coefficients.

Key Concepts

General Derivative Formula

Differentiation is a fundamental concept in calculus, allowing us to understand how functions change.
When addressing higher-order derivatives, the general derivative formula becomes a vital tool. The general derivative formula provides a way to express the result of taking the derivative of a function multiple times.

• It involves consistent patterns in how coefficients and exponents change with each order of differentiation.
• Understanding the general derivative formula makes it possible to find any nth derivative without manually recalculating every step from the beginning.
• This formula is particularly useful for functions with predictable differentiation behavior, such as polynomials and power functions.

In the exercise, the function given was a power function, \( x^{-1} \). By observing derivatives of increasing order, a pattern emerges. This pattern leads to the creation of the general formula, helping us efficiently compute derivatives of any order.

Factorial Sequence

A factorial sequence plays a significant role in generalizing derivative results, especially when dealing with orders of differentiation.
The factorial of a number \( n \) is the product of all positive integers up to \( n \). Notationally, it is represented as \( n! \). When looking at derivatives, factorial sequences often describe how coefficients evolve as we take higher-order derivatives.

• For example, the sequence in the nth derivative of the function \( x^{-1} \) follows values like -1, 2, -6, etc.
• These numbers are not random; they are rooted in factorial products and alternating signs.
• More precisely, the coefficients follow the pattern \((-1)^{n-1}(n-1)!\), indicating that factorials determine the size of the coefficients.

The factorial sequence maintains the structure of the coefficients, explaining their growth as the order of differentiation increases. It highlights the relationships among consecutive derivatives, revealing the systematic nature of differentiation.

Pattern Recognition in Derivatives

Identifying patterns in derivatives is essential for deriving general formulas.
This exercise requires recognizing the changing structure of terms as each derivative is computed.Pattern recognition allows us to:

• Predict the outcome of successive derivatives without computing each intermediary stage.
• Understand the changes in both the signs and magnitudes of coefficients in each term.
• Relate complex mathematical results to simple logic-based sequences.

The pattern in the exercise showed a sequence with alternating signs of coefficients. This alternation can be explained by the factor of \((-1)^n\), ensuring the signs change predictably.
Recognizing the pattern: - Enables the derivation of a general formula for any derivative order.- Simplifies complex differentiation problems through straightforward rules of transformation. Ultimately, pattern recognition is the backbone of simplifying and solving derivative problems effectively, especially in calculus differentiation.