91Ó°ÊÓ

The Power Rule is a fundamental concept in calculus that simplifies the process of finding the derivative of polynomial functions. When we have a function in the form of \( f(x) = x^n \), the Power Rule tells us that the derivative is \( f'(x) = n \cdot x^{n-1} \).

This means you multiply the variable's exponent by the coefficient and reduce the exponent by one. It's a quick and reliable way to find derivatives, especially when dealing with terms like \( x^3 \) or \( x^{-2} \).

• Start by identifying the exponent of the term.
• Apply the rule: multiply by the exponent.
• Decrease the exponent by one and adjust the term.

In the given problem, we had the function \( l(x) = \frac{1}{x^2} \), which we rewrote as \( l(x) = x^{-2} \). Applying the Power Rule here:

• We determine \( n = -2 \).
• The derivative then becomes \( l'(x) = -2 \cdot x^{-3} = -\frac{2}{x^3} \).

This solution highlights the simplicity and effectiveness of the Power Rule for functions expressed as exponents.

The Difference Quotient is a crucial concept that represents the average rate of change of a function over an interval. More formally, it's the expression used to define the derivative:

\[ \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} \]
This formula is foundational in calculus because it leads to the derivative by letting the interval \( h \) approach zero.

In this exercise, we applied the Difference Quotient to verify the derivative found using the Power Rule:

• We substituted \( l(x) = x^{-2} \) into the difference quotient.
• Simplifications were made leading to the expression \( \lim_{{h \to 0}} \frac{(x+h)^{-2} - x^{-2}}{h} \).
• Next, algebraic manipulations helped reduce this to \( \lim_{{h \to 0}} \frac{-2x-h}{x^2(x+h)^2} \).
• Finally, substituting \( h = 0 \), we obtained the confirmed derivative, \( -\frac{2}{x^3} \).

The Difference Quotient not only finds the derivative independently, but also corroborates methods like the Power Rule. Its thorough step-by-step approach builds a deeper understanding of how functions change, ensuring accuracy in calculus.

The Limit Definition in calculus is the underlying principle for understanding derivatives. A limit represents the value that a function approaches as the input approaches a certain point. In terms of derivatives, the limit is used to define the concept of the derivative at a specific point.

The formula is:
\[ \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} \]
This formula captures the idea of a tangent line to a curve. As \( h \to 0 \), the secant line approaches the tangent line, giving us the derivative, which is the instantaneous rate of change.

Using this for the function \( l(x) = \frac{1}{x^2} \), we:

• Expanded the expression to \( \lim_{{h \to 0}} \frac{1/(x+h)^2 - 1/x^2}{h} \).
• Performed algebraic simplifications and factored where possible.
• Evaluated the resulting simplified expression as \( h \to 0 \).
• This process confirmed \( l'(x) = -\frac{2}{x^3} \).

The limit definition is essential because it provides a rigorous foundation for derivatives, ensuring that any derivative found accurately reflects the function's behavior as inputs vary closely around a given point.