91Ó°ÊÓ

The power rule is a fundamental shortcut in calculus for finding the derivative of power functions. This rule states that for any function of the form \( f(x) = x^n \), the derivative is given by \( f^{\backslashprime}(x) = n \cdot x^{n-1} \).
For example, if \( f(x) = x^{3} \), then by applying the power rule, we get \( f^{\backslashprime}(x) = 3x^{2} \).
This method saves time and reduces reliance on the limit definition of derivatives. In our exercise, we applied the power rule to \( x^{-2} \), resulting in \( f^{\backslashprime}(x) = -2x^{-3} \).

The derivative of a function measures how the function's output changes as its input changes. It's akin to finding the slope of a function's graph at any point.
Mathematically, the derivative is defined as: \[ f^{\backslashprime}(x) = \backslashlim\backslashlimits_{h \backslashto 0} \backslasfrac{f(x+h) - f(x)}{h} \] However, for practical purposes, especially with polynomial functions, rules like the power rule simplify this process.
In our example, we started with \( f(x) = \backslashfrac{1}{x^{2}} \). By rewriting it as \( x^{-2} \), we easily applied the power rule to find its derivative. The final result was simplified to \( f^{\backslashprime}(x) = -\backslashfrac{2}{x^{3}} \), showing how quickly the function's value changes at any point x.

Calculus is a branch of mathematics focused on change. It includes two main areas: differentiation and integration. Differentiation deals with finding how a function changes at any given point. Integration, by contrast, accumulates values over an interval.
Derivatives are a key tool in calculus, enabling understanding of the rate of change, optimization problems, and motion analysis.
Rewriting and differentiating functions using rules like the power rule form the basics of calculus. For instance, transforming \( \backslashfrac{1}{x^{2}} \) to \( x^{-2} \) helps apply differentiation rules more efficiently. This exercise illustrates calculus's fundamental operations, showcasing its importance in mathematical analysis and everyday applications.