Differentials and their significance in calculus While being introduced to the concept of differentials, I did come across the fact that the differential of the function $f$ is not the function's absolute change. 
Mathematically, $$df=f'(x)dx\approx \Delta f$$
Given the function's analytical expression, why would one want to use differentials in computing change when $\Delta f$ always gives the true change no matter how small the interval over which we are computing the change is? 
Are there other applications of differentials that make them essential in calculus? 
 A: We should note that the near equality $df \approx \Delta f$ is true only on the condition that we've set $dx = \Delta x.$  The fundamental concept of differential calculus is that straight lines can approximate well-behaved curves.  ("Well-behaved" here means "can be approximated by straight lines", by the way.)  When we use $df \approx \Delta f$, we're approximating a curvy thing with a straight line.  We do this because it's usually easier to compute along straight lines than along curves.  See, for instance, the path-of-the-hurricane maps on the TV weather.  They are piece-wise bits of straight lines, because we can't compute along the curves as fast as the weather can.
A: One of the biggest overarching themes in geometry is that it's much easier to understand what happens at a point, than it is what happens over a larger region.
Just remembering the value of a function at a point, however, isn't enough information to do calculus; additionally remembering the differential at that point, however, lets you do a lot of useful things.
For example, you can compute $\mathrm{d}(f(x) g(x))$ using only the quantities attached to the point $x$: it is $f(x) \mathrm{d} g(x) + g(x) \mathrm{d}f(x)$.
Now, there is a theory of differences which parallels that of differentials. However, despite seeming like it's a more basic thing, it's a lot more complicated to use.
For example, if I pick some $\epsilon$ and define $\Delta f(x)$ to mean $f(x + \epsilon) - f(x)$, then we have either of the three formulas
$$\begin{align} \Delta(f(x) g(x)) &= f(x) \Delta g(x) + g(x + \epsilon) \Delta f(x) 
\\ &= f(x + \epsilon) \Delta g(x) + g(x) \Delta f(x)
\\ &= f(x) \Delta g(x) + g(x) \Delta f(x) + \Delta f(x) \Delta g(x)
\end{align}$$
Either way, you can see the product rule for differences is more complicated than the product rule for differentials. And things keep getting more and more complicated as you develop the theory. And even that's with a fixed $\epsilon$; it can become a nightmare if you let $\epsilon$ vary too.
