Given a function $f=f(x)$, the differential of $f$ is often defined to be \begin{equation} df = {df\over dx} \,dx \end{equation} We can think of $dx$ as an infinitesimal change in the $x$ direction, and $df$ as the corresponding infinitesimal change in $f$. Then the above equation shows how these infinitesimal changes are related. That's exactly what a derivative is, a ratio of infinitesimal changes!

One useful way to view this is to zoom in on the graph of $f$. If $f$ is differentiable, then if one zooms in far enough it will look straight, and the ratio of $\Delta f$ to $\Delta x$ will be indistinguishable from the derivative. We can think of $df$ as being the change in $f$ when we have zoomed in far enough that the error in this process is so small as to be irrelevant.

There are several ways to make this process precise, including the use of limits with which you are probably familiar. Another approach occurs in the branch of mathematics known as differential geometry, where $dx$ and $df$ are differential forms.

Another informal definition of differentials is that $dx$ and $df$ are things which you integrate. In fact, you are already used to using differentials this way — just think about substituting $u=x^2$ into the following integral: \begin{equation} \int 2x\sin(x^2)\,dx = \int \sin(u)\,du \end{equation} Didn't you use the relation $du = 2x\,dx$? Those are differentials!

Another way to think of differentials is as the numerators of derivatives, in the sense that “dividing” differentials leads to derivatives. For instance, if $u=x^2$ as above, and $x=\cos\theta$, then \begin{equation} du = 2x\,dx = 2x (-\sin\theta\,d\theta) = -2\cos\theta\sin\theta\,d\theta \end{equation} and “dividing” through by $d\theta$ yields \begin{equation} {du\over d\theta} = -2\cos\theta\sin\theta \end{equation} which is just the chain rule, since $u=x^2=\cos^2\theta$. In fact, using differentials makes the chain rule automatic!

Although it is hard to give a rigorous definition of differentials, these two examples should convince you that they are a useful tool. Even if you view the use of differentials as nothing more than formal manipulation, you will always get correct answers — provided, of course, that you manipulate them correctly. Note that differentials themselves are rarely the answer to any question — it's their ratios which matter.

What about functions of several variables? Suppose that $f=f(x,y,z)$. Then small changes in, say, $x$ will produce small changes in $f$, and similarly for small changes in $y$ or $z$. But changing one variable while holding the others fixed is precisely what partial differentiation is all about, so it should come as no surprise that the differential of $f$ is now \begin{equation} df = \Partial{f}{x}\,dx + \Partial{f}{y}\,dy + \Partial{f}{z}\,dz \end{equation}

For functions of two variables, $f=f(x,y)$, this relation can be interpreted in terms of the tangent plane to the graph of $z=f(x,y)$, whose equation takes the form \begin{equation} \Delta z = \Partial{f}{x}\,\Delta x + \Partial{f}{y}\,\Delta y \end{equation} This equation is linear; planes are straight. This is again just zooming in on the graph of $f$! But it takes two slopes to describe a plane, which are given by the partial derivatives of the function being graphed.

Once again, the use of differentials makes the chain rule automatic. Suppose that $x$, $y$, $z$ are functions of $u$. Then $dx={dx\over du}\,du$, etc.; substituting this into the above expression for $df$ and “dividing” by $du$ yields the standard chain rule formula for ${df\over du}$, namely \begin{equation} {df\over du} = \Partial{f}{x}{dx \over du} + \Partial{f}{y}{dy \over du} + \Partial{f}{z}{dz \over du} \end{equation} More complicated cases, such as $x$, $y$, $z$ being themselves functions of several variables, are only slightly more difficult.

First-year calculus courses often emphasize that derivatives are slopes, and integrals are areas. These interpretations are only sometimes correct, and in fact can cause problems for students in multivariable calculus. We prefer to make differentials fundamental.

     Derivatives are rates of change, sometimes slopes.

Derivatives describe how one physical quantity changes with respect to another. We therefore eschew the notation $f'(x)$, in favor of $\frac{df}{dx}$; it is essential to keep track of which quantity is changing. Derivatives can therefore be thought of as ratios of differentials

     Integrals are sums, sometimes areas.

Integrals describe the total amount of some physical quantity, and involve chopping up a region into small pieces, then adding up the quantity on all the pieces. It is important to keep track of what one is adding, and where. Many students have difficulty with wording like
     “the part of the paraboloid $z=r^2$ which lies inside the cylinder $r=3$”
not realizing that the surface is a paraboloid, not a cylinder — presumably because they mix up “what” and “where”. 1) From this point of view, the differential(s) in the integrand are part of “what”; we abhor the notation, common among students (and included in some textbooks!), “$\int\!f\,$” for the antiderivative of $f$.