# Definition:Differential of Mapping/Warning

## Warning

It is generally considered to be **incorrect** to consider $\d y$ as:

*a small change in $y$ caused by a small change $\d x$ in $x$.*

This is *nearly* true for small values of $\d x$, but will only ever be *exactly* true when $f$ has a graph which is a straight line.

If it is necessary to talk about *small changes* then the notation $\delta x$ and $\delta y$ are to be used instead.

Thus:

- $\ds \lim_{\delta x \mathop \to 0} \delta y = \frac {\d y} {\d x} \delta x$

Received wisdom tells us that an even worse misconception is the idea that $\d y$ and $\d x$ are infinitesimal quantities which are obtained by letting $\delta x$ and $\delta y$ tend to zero.

Then $\dfrac {\d y} {\d x}$ could be regarded as the quotient of these quantities, and the whole concept of a limit could be disposed of. This was the original idea that Isaac Newton based his *Theory of Fluxions* on. However, useful as this approach is, it is generally considered that does not have any logical basis.

However, the field of nonstandard analysis is an attempt to address these concerns from a modern perspective.