What are differentials?

I don't know what is going on at all.

I am supposed to find the differential of each function. It was explained thoroughly to me for about 2 hours but I can't figure it out. Stewart is amazing at doing that. The first problem he wants me to solve sure is easy: $y=e^{x/10}$, $x=0$, $dx=.1$. He gave me a formula of $dy=f'(x) \, dx$ I don't really know what that is but it doesn't seem to work when I try to use it.

-
See also this previous question; it may help. – Arturo Magidin Oct 4 '11 at 16:09

It depends on which course are you following, but the most basic explanation. Given a differentiable function $f$ on the real line, a point $x$ and an increment $h$ you would like to find how will the value of $f$ change from $f(x)$ to $f(x+h)$. Clearly, the good way to compare them is to subtract: $$\Delta f(x,h) = f(x+h) - f(x).$$ For example, if $f = kx$ then $\Delta f(x,h) = k(x+h)-kx = kh$.
If $f$ has more complicated form, the structure of its increment $\Delta f(x,h)$ could be weird. On the other hand, if the increment $h$ is small - you can still expect kind linear behavior of $\Delta f$. You decompose $\Delta f$ into two terms - the main (linear) one and the rest which should be very 'small': $$\Delta f(x,h) = k(x)\cdot h+\varepsilon(h)h\quad(\star)$$ where $\varepsilon(h)\to 0$ with $h\to 0$, so it is negligible. How to find such a value $k(x)$ at $x$? Ok, we know that $\varepsilon(h)\to 0$ so we should just extract $\varepsilon(h)$ from $(\star)$ and calculate the limit: $$\varepsilon(h) = \frac{\Delta f(x,h)}{h}-k\to0,$$ so $$k = \lim\limits_{h\to 0}\frac{\Delta f(x,h)}{h} = \lim\limits_{h\to 0}\frac{f(x+h)-f(x)}{h} = f'(x).$$ The part $k(x)\cdot h = f'(x)h=:\mathrm df(x,h)$ is called the differential of $f$ at the point $x$ and increment $h$. Note that $\mathrm df(x,h)$ is a function both of a point and the increment.
Now about your example: $f'(x) = \frac1{10}e^{x/10}|_{x=0} = \frac1{10}$. The increment denoted as $\mathrm dx (= h) = 0.1$, so $$\mathrm df(x,dx) = \frac1{10}\cdot 0.1 = \frac1{100}.$$
Gortaur explains pretty well how to construct your answer using the definition of the derivative. You can also use the rules of differentiation (which are just shortcuts to finding the above). In this case, you would need to know that the derivative of $e^x$ is $e^x$ and that the derivative of $f(g(x))$ is $f'(g(x))g'(x)$ (the chain rule). – kbolino Oct 4 '11 at 18:56