that the two points on the curve with y-values $y(x)$ and $y(x+dx))$ are just a dibbl away from each other.

I have already said that a dibbl is a quantity so small that we can take its square to be zero “for all practical purposes.” So the rest of this article makes use of “the dibbl equation” $$dx \times dx = dx^{2} = 0
\label{eq:dibbl}$$

The dibbl equation comes into play when we recalculate the slope of a quadratic at the point $y(x)$. We write, for $y(x)=cx^{2}$: $$\begin{aligned}
\mbox{slope} &= &\frac{y(x+dx)-y(x)}{(x+dx)-x} \label{eq:q3}\\
 & = &\frac{c(x^{2}+2cxdx +  \overbrace{dx^{2}}^{=0})-cx^{2}}{dx} \nonumber \\
 &= &\frac{2cx\times dx \hspace{-2ex}//}{dx \hspace{-2ex}//} = 2cx  \nonumber\end{aligned}$$

Eq. \[eq:q3\] no longer requires the step of setting the point $x_{2}$ equal to the point $x_{1}$. The dibbl equation makes that step unnecessary.

Note that the slope in Eq. \[eq:q3\] is a function of $x$. It is, in fact, a function of $x$ that was [*[derived]{}*]{} from the original function $y =
cx^{2}$. Mathematicians, for this reason, call the slope of a function y(x) the [*derivative*]{} and write it as a ratio of two dibbls, dy and dx. To summarize, then, for any smooth function $y(x)$ there is a slope function, called 