Given a function $f=f(x)$, the differential of $f$ is defined to be $$df = {df\over dx} \,dx$$ This definition is not very satisfying. What, after all, is $dx$? There are several ways to make this precise. In the branch of mathematics known as differential geometry, $dx$ and $df$ are differential forms, but this is more sophistication than we need.

We can safely 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 can also regard $dx$ and $df$ as “things which you integrate”. You are already used to using differentials this way — just think about substituting $u=x^2$ into the following integral: $$\int 2x\sin(x^2)\,dx = \int \sin(u)\,du$$ 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 $$du = 2x\,dx = 2x (-\sin\theta\,d\theta) = -2\cos\theta\sin\theta\,d\theta$$ and “dividing” through by $d\theta$ yields $${du\over d\theta} = -2\cos\theta\sin\theta$$ 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 at this mathematical level, 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.

What about a small changes in a function of two variables, $f=f(x,y)$? We can approximate this change using the tangent plane to the graph of $z=f(x,y)$ which can be written in the form $$\Delta z = \Partial{f}{x}\,\Delta x + \Partial{f}{y}\,\Delta y$$ where the partial derivatives are to be evaluated at the point where the tangent plane touches the graph. This equation is linear; planes are straight. And if you know how a plane is tilted in two directions, you know everything about it — it takes precisely two slopes, given by the partial derivatives, to determine the plane.

So we have $$\Delta f \approx \Delta z$$ with $\Delta z$ as above. As we zoom in, the approximation becomes better and better, so we write $$df = \Partial{f}{x}\,dx + \Partial{f}{y}\,dy$$

A similar argument works for functions of 3 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 $$df = \Partial{f}{x}\,dx + \Partial{f}{y}\,dy + \Partial{f}{z}\,dz$$

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 $${df\over du} = \Partial{f}{x}{dx \over du} + \Partial{f}{y}{dy \over du} + \Partial{f}{z}{dz \over du}$$ More complicated cases, such as $x$, $y$, $z$ being themselves functions of several variables, are only slightly more difficult once you understand the basic idea.

GOALS