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## Section4.1Definition 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 $dx\cong0\text{.}$