# Definition:Differential of Mapping/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 $\map {\d f} x : \R \to \R$ defined as:

- $\map {\map {\d f} x} h = \map {f'} x \cdot h$

where $\map {f'} x$ 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** $\d f$ is the mapping $\d f : U \to \map {\operatorname {Hom} } {\R, \R}$ defined as:

- $\map {\paren {\d f} } x = \map {\d f} x$

where:

- $\map {\d f} x$ is the differential of $f$ at $x$
- $\map {\operatorname {Hom} } {\R, \R}$ is the set of all linear transformations from $\R$ to $\R$.

Also:

- $\map f {x + h} - \map f x - \map {\d f} {x; h} = \map o h$

as $h \to 0$.

In the above, $\map o h$ is interpreted as little-O of $h$.

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## Notation

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

- $\map {\d f} x$
- $\d f_x$
- $\d_x f$
- $\map {D f} x$
- $D_x f$

Substituting $\d y$ for $\map {\d f} {x; h}$ and $\d x$ for $h$, the following notation emerges:

- $\d y = \map {f'} x \rd x$

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:

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

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 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:

- $k = \map {\d f} {x; h} = \map {f'} x h$

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): $\S 10.4$