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Section 4.1 Definition of the Derivative

The derivative of a function \(f(x)\) is the function that gives its slope at each point. To find the slope, we determine the slope of the secant line connecting two nearby points on the graph \(y=f(x)\text{.}\) Using infinitesimals, we can choose these points to be “close”!

We therefore define the derivative \(f'(a)\) of \(f(x)\) at any point \(x=a\) with \(a\) real to be

\begin{equation} f'(a) \cong \frac{f(a+dx)-f(a)}{dx}\tag{4.1.1} \end{equation}

which is again a real number.  1 We have used the hyperreal numbers to define the derivative as a real function. But it is straightforward to extend real functions to hyperreal functions, as guaranteed by the transfer principle. The derivative is only well-defined if the result of this computation is the same for every infinitesimal \(dx\text{.}\)

This definition has two parts. First, a traditional difference quotient is constructed, with the “smallness” of \(dx\) being captured by requiring it to be infinitesimal (and hence nonzero!). Then, the resulting hyperreal number is converted to the real number \(f'(a)\) by taking the standard part. The traditional notion of limits is therefore neatly avoided.

Consider for example the function \(f(x)=x^2\text{.}\) From the definition, we have

\begin{align} \frac{(a+dx)^2-a^2}{dx} \amp= \frac{2a\,dx+dx^2}{dx}\notag\\ \amp= 2a + dx \cong 2a\tag{4.1.2} \end{align}

so that \(f'(a)=2a\text{,}\) and the derivative of \(x^2\) with respect to \(x\) is \(2x\) as expected. This computation is valid for any infinitesimal \(dx\text{,}\) since \(dx\cong0\text{.}\)