TITLE: What is the practical difference between a differential and a derivative?
QUESTION [264 upvotes]: I ask because, as a first-year calculus student, I am running into the fact that I didn't quite get this down when understanding the derivative:
So, a derivative is the rate of change of a function with respect to changes in its variable, this much I get.
Thing is, definitions of 'differential' tend to be in the form of defining the derivative and calling the differential 'an infinitesimally small change in x', which is fine as far it goes, but then why bother even defining it formally outside of needing it for derivatives?
And THEN, the bloody differential starts showing up as a function in integrals, where it appears to be ignored part of the time, then functioning as a variable the rest.
Why do I say 'practical'? Because when I asked for an explanation from other mathematician parties, I got one involving the graph of the function and how, given a right-angle triangle, a derivative is one of the other angles, where the differential is the line opposite the angle. 
I'm sure that explanation is correct as far it goes, but it doesn't tell me what the differential DOES, or why it's useful, which are the two facts I need in order to really understand it.
Any assistance?

REPLY [5 votes]: These answers haven't formalized the objects $dx$, so I'll give my own answer which does. This will be more high level and requires some understanding of linear algebra, group actions, and calculus in more than one variable. There's a few objects we need to make clear: the basis vector $\frac{\partial}{\partial x}$, the projection map $x^i$, and the operator d. 
The setup for our story is going to be in affine space $\mathbb{A}^n$ which is defined as $\mathbb{R}^n$ with a simply transitive group action on itself by translations. We say that $\mathbb{A}^n$ is defined over $\mathbb{R}^n$ and think of elements in the former as points and the elements in the latter as vectors. The upshot of the words "simply transitive" is that subtracting two points always gives a unique vector. If these words are unfamiliar, feel free to think of $\mathbb{A}^n$ as $\mathbb{R}^n$ + translations. We define an affine subspace $A\subset \mathbb{A}^n$ to be a subset with a simply transitive group action from a linear subspace $V\subset \mathbb{R}^n$. We call $V$ the tangent space of $A$. Note that $\mathbb{R}^n$ comes with canonically equipped with a set of basis vectors $\{e_i\}$ with  $e_i=(0,...,1,...0)$ where the 1 is in the $i$th position. We rename $\frac{\partial}{\partial x^i}:=e_i$. We also have the standard projection maps $x^i:\mathbb{R}^n\to \mathbb{R}$ by remembering only the $i$th coordinate. We define $dx^i\in (\mathbb{R}^n)^* $ to be the dual to vector $\frac{\partial}{\partial x^i}$, so $\{dx^i\}$ forms a basis for $(\mathbb{R}^n)^*$. Since we built $\mathbb{A}^n$ on top of $\mathbb{R}^n$, we also get a canonical basis and projection maps on $\mathbb{A}^n$.
For example, fix $\mathbb{A}^2$ and let $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$ denote the basis vector for the underlying tangent space $\mathbb{R}^2$. The graph of equation $y=3$ is an affine subspace of $\mathbb{A}^2$ with tangent space the span of $\frac{\partial}{\partial x}\in \mathbb{R}^2$. Another example would be the graph of the equation $y=x$  is an affine subspace with tangent space the span of $\frac{\partial}{\partial x}+\frac{\partial}{\partial y}$. 
At this point in our story we require that $\mathbb{R}^n$ has a norm denoted $\|\cdot\|$ which induces a distance function on $\mathbb{A}^n$ via subtraction. Let $U\subset \mathbb{A}^n$ be a open set, and $p\in U$ a point. We say $f:U\to \mathbb{A}^m$ is differentiable at $p$ if there exists a linear function which we denote $df_p:\mathbb{R}^n\to \mathbb{R}^m$ which satisfies the following inequality: For every $\epsilon>0$, there exists a $\delta$ such that if $\|\xi\|<\delta$, then
$$\|f(p+\xi)-f(p)-df_p(\xi)\|<\epsilon \|\xi\|$$
We say $f$ is differentiable on $U$ if it's differentiable at every $p\in U$. One may  call $df_p$ the derivative at $p$ in a multivariable class. Fix codomain $\mathbb{A}^1$. We don't lose generality by restricting the codomain to one dimensions becuase every function $f:\mathbb{A}^n\to \mathbb{A}^m$ can be expressed as coordinate functions $f=(f^1,...,f^n)$ by composing with projection map $x^i$. Let me denote Diff($U$,$\mathbb{A}^1$) to be the set of differentiable functions between the two, and Map($U$,$\mathbb{A}^1$) the set of all functions between the two. These two form a vector space structure by function addition and scalar multiplication defined in the usual way. We define an operator between the these two vector spaces
$$\mathfrak{d}:\text{Diff}(U,\mathbb{A}^1)\to \text{Map}(U,\mathbb{A}^1)$$
We require $\mathfrak{d}$ to be linear and obey the Leibniz rule, namely given $f,g\in \text{Diff}(U,\mathbb{A}^1)$
$$\mathfrak{d}(fg)=\mathfrak{d}f \cdot g+ f\cdot \mathfrak{d}f$$
We call $\mathfrak{d}$ a derivation if it satisfies the above conditions, and we see that the differential operator $d$ is a special case of this. So it turns out that taking $U=\mathbb{A}^n$ and $f=x^i$ the projection map, $dx^i$ which we defined as the dual basis to $\frac{d}{dx^i}$ is actually the differential of $x^i$ (I'll leave this to whoever the hell read this far to show). There's a whole lot more to be said and skimmed over. If you're interested, all this information and more comes from here.