Definition:Differential of Mapping/Real Function

Definition
Let $f: \R \to \R$ be a real function which is differentiable at a point $x \in \R$.

The differential $\mathrm d f$ can be regarded as a (real) function of two variables, defined as:
 * $\mathrm d f \left({x; h}\right) = f' \left({x}\right) h$

where $f' \left({x}\right)$ is the derivative of $f$ at $x$.

Also:
 * $f \left({x + h}\right) - f \left({x}\right) - \mathrm d f \left({x; h}\right) = o \left({h}\right)$

as $h \to 0$.

In the above, $o \left({h}\right)$ is interpreted as little-O of $h$.

Notation
The differential of $f$ at $x$ can also be denoted $\mathrm d_x f$ or $\mathrm d f_x$.

Substituting $\mathrm d y$ for $\mathrm d f \left({x; h}\right)$ and $\mathrm d x$ for $h$, the following notation emerges:
 * $\mathrm d y = f' \left({x}\right) \mathrm d x$

hence:
 * $\mathrm d y = \dfrac {\mathrm d y} {\mathrm d x} \mathrm d x$

Warning
It is false to consider $\mathrm d y$ as:
 * a small change in $y$ caused by a small change $\mathrm d x$ in $x$.

This is nearly true for small values of $\mathrm d x$, 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:
 * $\displaystyle \lim_{\delta x \to 0} \ \delta y = \frac {\mathrm d y} {\mathrm d x} \delta x$

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

Then $\dfrac {\mathrm d y} {\mathrm d x}$ 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 non-standard analysis is an attempt to address these concerns from a modern perspective.

Also see

 * Straight Line Defined by Differential, where it is shown that for any fixed $x \in \R$, the equation:
 * $k = \mathrm d f \left({x; h}\right) = f' \left({x}\right) h$

is the equation of a straight line, tangent to the graph of the real function $f$ at the point $x$.