Definition:Differential of Mapping

Real Valued Functions
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 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 $x$.

In the literature the differential of $f$ at $x$ is sometimes denoted by $\mathrm d_x f$ or $\mathrm d f_x$.

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

More generally, if $f : \R^n \to \R$, then the differential of $f$ at $x = (x_1,\ldots,x_n) \in \R^n$ is defined by:
 * $\displaystyle\mathrm d f(x,h) = \frac{\partial f}{\partial x_1}h_1 + \cdots + \frac{\partial f}{\partial x_n}h_n$

where $h = (h_1,\ldots,h_n) \in \R^n$.

As before we find that $\mathrm d x_i(x,h) = h_i$ for $i = 1,\ldots,n$, so
 * $\displaystyle\mathrm d f = \frac{\partial f}{\partial x_1}\mathrm d x_1 + \cdots + \frac{\partial f}{\partial x_n}\mathrm d x_n$

Vector Spaces
Let $V$ and $W$ be finite dimensional vector spaces over $\R$.

Let $O \subseteq V$ be an open set.

Let $f : O \to W$ be a function differentiable at each point of $U$.

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

where $D_x f$ is the derivative of $f$ at the point $x \in U$, a bounded linear operator from $V$ to $W$.

Interpretation in co-ordinates
Let the dimensions of $V$ and $W$ be $n$ and $m$ respectively.

By Vector Spaces of the same Dimension are Isomorphic, we identify points in $V$ with $n$-tuples $(x_1,\ldots,x_n)$ of real numbers.

By Vector Valued Function is Differentiable iff each Coordinate is Differentiable, there are differentiable functions $f_1,\ldots,f_m : \R^n \to \R$ such that for all $x \in \R^n$
 * $f(x) =

\begin{pmatrix} f_1(x)\\ \vdots\\ f_m(x) \end{pmatrix}$

Using the additional fact that derivative of function is Jacobian, we know that for $h = (h_1,\ldots,h_n) \in \R^n$ the differential at $x \in O$ is given by
 * $\displaystyle \mathrm d f(x,h) = (D_x f) (h)

\begin{pmatrix} \frac{\partial f_1}{\partial x_1} & \cdots & \frac{\partial f_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1} & \cdots & \frac{\partial f_m}{\partial x_n} \end{pmatrix} \begin{pmatrix} h_1\\ \vdots\\ h_n \end{pmatrix}$

Consider now the case $m = 1$.

We can regard each variable $x_i$ as the identity function on the $i^\text{th}$ co-ordinate, and zero on all other co-ordinates.

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

Thus, omitting the arguments, the differential can be written:
 * $\mathrm d f = \dfrac{\partial f}{\partial x_1} \mathrm d x_1 + \cdots +  \dfrac{\partial f}{\partial x_n} \mathrm d x_n$

Moreover, when $n = 1$ also,


 * $\mathrm d f = f' \left({x}\right) \mathrm d x$

and we recover the definitions given for functions $\R^n \to \R$ above.

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.