# Definition:Differential/Real Function

## Definition

### At a Point

Let $U \subset \R$ be an open set.

Let $f: U \to \R$ be a real function.

Let $f$ be differentiable at a point $x \in U$.

The **differential of $f$ at $x$** is the linear transformation $\rd f \left({x}\right) : \R \to \R$ defined as:

- $\rd f \left({x}\right) \left({h}\right) = f' \left({x}\right) \cdot h$

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

### On an Open Set

Let $U \subset \R$ be an open set.

Let $f : U \to \R$ be a real function.

Let $f$ be differentiable in $U$.

The **differential** $\rd f$ is the mapping $\rd f : U \to \operatorname{Hom} \left({\R, \R}\right)$ defined as:

- $\left({\mathrm d f}\right) \left({x}\right) = \rd f \left({x}\right)$

where:

- $\rd f \left({x}\right)$ is the differential of $f$ at $x$
- $\operatorname{Hom} \left({\R, \R}\right)$ is the set of all linear transformations from $\R$ to $\R$.

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

There are various notations for the differential of a function $f$ at $x$:

- $\d f \left({x}\right)$
- $\d f_x$
- $\d_x f$
- $D f \left({x}\right)$
- $D_x f$

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

- $\d y = f' \left({x}\right) \rd x$

hence:

- $\d y = \dfrac {\d y} {\d x} \rd x$

## 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:

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

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 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 $x$.

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 13$: Differentials: $13.49$, $13.50$ - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 10.4$