Index notation and total differential I understand that the gradient $\partial_i$ is covariant.
Let f be a function of 3 variables
So I can write the total differential as
$$
df=\partial_1fdx^1+\partial_2fdx^2+\partial_3fdx^3 = \partial_kfdx^k,
$$
summing correctly over the same lower and upper index.
But when I write
$
\displaystyle\partial_i=\frac{\partial}{\partial x^i}
$ (and this is how it is written in all books)
and want to write the total differential, I have to write
$$
df= \frac{\partial f}{\partial x^1}dx^1+\frac{\partial f}{\partial x^2}dx^2+\frac{\partial f}{\partial x^3}dx^3=\frac{\partial f}{\partial x^k}dx^k
$$
and so I am summing over the same upper indices.
My question: Why does an index get raised when it is moved to the denominator?
 A: A raised index in the denominator counts as a lowered index (because it is in the denominator)
See "Introduction to Smooth Manifolds" by J. Lee. I believe it is in both editions but in the second edition see Chapter 3 on tangent vectors and look at the text below Eq. 3.2:

$$D_v |_a f = v^i \frac{\partial f}{\partial x^i}(a)$$
(Here we are using the summation convention as usual, so the expression on the right-hand side is understood to be summed over $i=1, \ldots, n$. This sum is consistent with our index convention if we stipulate that an upper index "in the denominator" is to be regarded as a lower index.)

Here $D_v|_a$ is the derivative of function $f$ in the direction of vector $v = v^i e_i|_a$ at point $a$ where $v^i$ are the components of $v$ with respect to the basis $e_i|_a$, where the basis $e_i|_a$ is defined at point $a$.
Lee goes on to introduce the concept of derivations and how they can be used to generally define tangent vectors on manifolds. It turns out that the map:
$$
\frac{\partial}{\partial x^i}\big|_a
$$
defined by
$$
\frac{\partial}{\partial x^i}\big|_a\left(f\right) = \frac{\partial f}{\partial x^i}(a)
$$
is a derivation and can be thought of as a tangent vector. Since we agreed above that something like $\frac{\partial}{\partial x^i}$ is to be thought of as a `lower index thing' we can equate it with another lower index thing and define
$$
\partial_i = \frac{\partial}{\partial x^i}
$$
A: $\partial_i$ is defined as $\dfrac{\partial}{\partial x^i}$. Therefore the index does not get raised when brought to the denominator, rather it is notation for the derivative justified rigorously:
In Cartesian coordinates the partial derivative transforms as a covariant vector
$$\frac{\partial}{\partial \bar{x}^i}=\frac{\partial x^j}{\partial \bar{x}^i}\frac{\partial}{\partial x^j}$$
Covariant vectors are written with lowered indices.
