Definition:Differential of Mapping

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

The differential $\mathrm d f$ can be regarded as a function of two variables, defined as:
 * $\mathrm d f \left({x; h}\right) = f' \left({x}\right) h \qquad (1)$

where $f' \left({x}\right)$ is the derivative of $f$ at $x$, and:
 * $f \left({x + h}\right) - f \left({x}\right) - \mathrm d f \left({x; h}\right) = o \left({h}\right)$

as $h \to 0$.

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

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.

Now we can regard the variable $x$ as the identity function from $\R$ to $\R$.

From the definition above and derivative of identity function we see that $\mathrm d x(x,h) = h$.

Moreover if we let $y = f\left({ x }\right)$, then trivially $\mathrm d y \left({x; h}\right) = \mathrm d f \left({x; h}\right)$, because we have only changed the notation. This shows that
 * $\mathrm d y = \mathrm d y \left({x; h}\right) = \mathrm d f \left({x; h}\right)$ and $\mathrm d x = \mathrm d x(x,h) = h$

Thus the formula $(1)$ can be written:
 * $\mathrm d y = f' \left({x}\right) \mathrm d x$

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 known as
In the literature the differential of $f$ at $x$ is sometimes denoted by $\mathrm d_x f$.

In this case we can view $\mathrm d f$ as a function from $\R$ into the cotangent space:
 * $\mathrm d f : x \mapsto \mathrm d_x f$

From the fact that the equation:
 * $k = \mathrm d f \left({x; h}\right) = f' \left({x}\right) h$

is the equation of a straight line (see above) the differential is sometimes called the linearization of $f$ at $x$.