Definition:Differential/Real Function

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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

$k = \mathrm d f \left({x; h}\right) = f' \left({x}\right) h$

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


Sources