91Ó°ÊÓ

The power rule is one of the fundamental tools in calculus for finding derivatives of functions that are expressed as powers of variables. It provides a quick and efficient way to differentiate functions of the form \( f(x) = x^n \). The rule states that if \( f(x) = x^n \), then the derivative \( f'(x) = nx^{n-1} \). This means you can multiply the exponent by the coefficient and then reduce the exponent by one.

In the original exercise, the power rule is applied to functions like \( f(x) = \frac{1}{x} \) and \( f(x) = \frac{1}{x^2} \). These functions are rewritten using negative exponents: \( f(x) = x^{-1} \) and \( f(x) = x^{-2} \).

Using the power rule:

• For \( f(x) = x^{-1} \), the first derivative is \( f'(x) = -1x^{-2} = -\frac{1}{x^2}\).
• For \( f(x) = x^{-2} \), the first derivative is \( f'(x) = -2x^{-3} = -\frac{2}{x^3}\).

Notice how the power rule simplifies the process of finding derivatives for any power of \( x \). This is a crucial step in not just solving derivatives but also in understanding how the function changes at any given point.

The "nth derivative" refers to taking the derivative of a function multiple times, which allows us to explore the deeper layers of how a function behaves.Finding the nth derivative involves identifying a pattern from the successive derivatives. The nth derivative \( f^{(n)}(x) \) will show how the higher-order behavior of the function can be generalized.

In the provided solution, after calculating the first few derivatives:

• For \( f(x) = \frac{1}{x} \): We found \( f'(x) = -\frac{1}{x^2} \), \( f''(x) = \frac{2}{x^3} \), and \( f'''(x) = -\frac{6}{x^4} \). The pattern leads to the formula \( f^{(n)}(x) = (-1)^n n! x^{-(n+1)} \).
• For \( f(x) = \frac{1}{x^2} \): We found \( f'(x) = -\frac{2}{x^3} \), \( f''(x) = \frac{6}{x^4} \), and \( f'''(x) = -\frac{24}{x^5} \). The pattern forms the formula \( f^{(n)}(x) = (-1)^n (n+1)! x^{-(n+2)} \).

This process of finding the nth derivative helps identify general patterns and properties of the function, making it easier to predict its behavior in different contexts.

Derivative formulas are standardized expressions that provide quick methods to calculate the derivative of various functions. These formulas include rules and patterns that apply to many types of functions and operations, such as the power rule, product rule, and chain rule.

In the context of the given exercise, we see the use of derivative formulas like:

• Power Rule: As described earlier, useful for functions of the form \( x^n \).
• Factorials in nth Derivatives: For functions such as \( 1/x \) or \( 1/x^2 \), the factorial (\( n! \)) plays a significant role when expressing nth derivatives. This is evident in the formulas derived from the patterns in the solution:
• Alternating Sign: The expression \((-1)^n\) is crucial for alternating the sign of each derivative, depending on whether \( n \) is odd or even. This is observed in the patterns: \( f^{(n)}(x) = (-1)^n n! x^{-(n+1)} \) and \( f^{(n)}(x) = (-1)^n (n+1)! x^{-(n+2)} \).

These patterns and rules streamline the differentiation process, making it quicker and more efficient, and showcase the elegance of mathematical structures in calculus.