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## Section4.3Power Rule

We can generalize the earlier example ($f(x)=x^2$) to arbitrary positive integer powers. Suppose $f(x)=x^n\text{.}$ From the definition, we have

\begin{align} \frac{(x+dx)^n-x^n}{dx} \amp= \frac{nx^{n-1}\,dx+\frac{n(n-1)}{2}x^{n-2}\,dx^2+...}{dx}\notag\\ \amp= nx^{n-1} + \frac{n(n-1)}{2}x^{n-2}\,dx + ... \cong nx^{n-1} .\label{dpd}\tag{4.3.1} \end{align}

Thus,

\begin{equation} f'(x) = nx^{n-1}\label{dpn}\tag{4.3.2} \end{equation}

or equivalently

\begin{equation} \frac{d(x^n)}{dx} = nx^{n-1}\text{.}\label{dpl}\tag{4.3.3} \end{equation}

This derivative rule can also be written as

\begin{equation} d(x^n) \approx nx^{n-1}\,dx \label{dpa}\tag{4.3.4} \end{equation}

which is however usually written as an equality, namely

\begin{equation} d(x^n) = nx^{n-1}\,dx\text{.}\label{dpe}\tag{4.3.5} \end{equation}

This derivation demonstrates the useful relationship between the hyperreals and the reals. The derivative (4.3.2) is initially constructed only for real values of $x\text{,}$ but the computation (4.3.1) is done over the hyperreals. The resulting expression for the derivative can, however, immediately be extended to the hyperreals if desired. The differential form (4.3.4) of the derivative rule, on the other hand, is a statement about closeness over the hyperreals, and has no exact analog over the reals. The common representation (4.3.5) can be interpreted over the reals as an assertion about linearization and/or approximation, or as a shorthand for the hyperreal assertion (4.3.4). We will henceforth adopt this latter interpretation, regarding all of (4.3.3)–(4.3.5) as equivalent.