$\int_a^b\frac{d}{dx}f(x,y)dy$ versus $\int_a^b\frac{\partial}{\partial x}f(x,y)dy$, which is correct? For a two-variable function $f$, which symbol below is technically correct?

$$ \int_a^b\frac{d}{dx}f(x,y)dy ~~~~~~~ \text{versus} ~~~~~~~ \int_a^b\frac{\partial}{\partial x}f(x,y)dy$$

The first or the second? Or both? How about $\int_a^b\frac{\partial f}{\partial x}(x,y)dy$?
Notice that the point here is that for a two-variable function, can't we use $d/dx$? Must we use $\partial/\partial x$? You may say, there's a rule that whenever dealing with multivariable function, use $\partial$. However, notice that you would write $\frac{df(x,3)}{dx}$ instead of $\frac{\partial f(x,3)}{\partial x}$. Here $f$ itself is a two-variable function, but we're not concerning $f$ itself, but the expression $f(x,3)$ (just as you must have seen something like $\frac{d(\sin x+x^2)}{dx}$, imagine that it is the case that the nominatior $(\sin x+x^2)$ is replaced by the expression $f(x,3)$). So the rule that "whenever dealing with multivariable function, use $\partial$" does not make sense.
And on the other hand, when INSIDE the integral sign, the variable $y$ is somewhat already served as a constant (we say the variable is being binded), waiting to be integrate after the integrand is completely evaluated (just as in $\sum_{i=1}^{10}i^2$, the variable $i$ inside summation is binded).  So, the all story happen inside the integral sign, is only of one variable $x$. In this way, should we write $\int_a^b\frac{d}{dx}f(x,y)dy$?
PS: Feel free to comment or to answer. Any advice or experience is welcomed. :)
 A: For $f : \Omega \subseteq \mathbb{R}^2 \to \mathbb{R} : (x,y) \mapsto f(x,y)$ it does not really matter if we write $\displaystyle \frac{df}{dx}$ or $\displaystyle \frac{\partial f}{\partial x}$ for the derivative with respect to $x$. It is a matter of emphasis. We usually write $\displaystyle \frac{df}{dx}$ to emphasize we are deriving a function of one variable, namely $x$, and $\displaystyle \frac{\partial f}{\partial x}$ to emphasize we are deriving a function of more than one variable with respect to the variable named $x$. It is a question of making easier to the reader to know what is going on. It is much more common to use $\displaystyle \frac{\partial}{\partial x}$ in this case.
Now regarding to the technicality, $\displaystyle \int_a^b\frac{d}{dx}f(x,y)\ dy$ and $\displaystyle \int_a^b\frac{\partial}{\partial x}f(x,y)\ dy$ and $\displaystyle \int_a^b\frac{\partial f}{\partial x}(x,y)\ dy$ are not correct or incorrect but inaccurate.
Derivatives acts on functions, not on points. That is $\displaystyle \frac{\partial}{\partial x}$ acts on the function $f$ and not on the point $f(x,y)$. Furthermore we derive the function $f$ at the point $(p,q)$ with respect to $x$, we not derive the function $f$ at the point $(x,y)$ with respect to $x$. That is
$$ \frac{\partial f}{\partial x} : \Omega \to \mathbb{R} : (p,q) \mapsto \frac{\partial f}{\partial x}(p,q) $$
The latter one, $\displaystyle \frac{\partial f}{\partial x}(x,y)$, creates unnecessary ambiguity. To see that, set $x = 1$; then $\displaystyle \frac{\partial f}{\partial x}(x,y) = \frac{\partial f}{\partial 1}(1,y)$ or $\displaystyle = \frac{\partial f}{\partial x}(1,y)$ or $\displaystyle = \frac{\partial f}{\partial 1}(x,y)$?
Therefere, agreeing we want to avoid ambiguity we would write
$$ \int_a^b\frac{\partial f}{\partial x}(p,q)\ dq $$
However it also reasonable to write
$$ \int_a^b\frac{\partial f}{\partial x}(p,y)\ dy $$
because the derivative are not binding the variable $y$ in any way.
Let me know if you need further clarifications.
A: The integrand is a function of two variables, $x$ and $y$, so use the partial derivative notation inside the integral sign.
$$\int\frac{\partial f(x,y)}{\partial x}\,dy$$
or
$$\int\frac{\partial }{\partial x}f(x,y)\,dy$$
are both fine.
The entire integral is a function of one variable, $x$, so use the total derivative notation outside the integral sign.
$$\frac{d}{dx}\int f(x,y)\,dy$$
A: In my experience, the most usual notation (and probably most correct) is : 
$$\int_a^b\frac{\partial f}{\partial x}(x,y)dy$$
To justify my saying :
$(1)$ : $f(x,y)$ is a $2$-variable function over $\mathbb R^2$ and hence the derivative with respect to one of its variables (let's say $x$) is notated as : 
$$f_x = \frac{\partial f}{\partial x}$$
$(2)$ : To note what variables/constants etc. the partial derivative is calculated over, one writes them in a parentheses following the notation mentioned above, so the partial derivative of the function $f(x,y)$ with respect to $x$ involving the variables $(x,y)$ (without it being specified over certain $x_0,y_0$) is :  
$$f_x = \frac{\partial f}{\partial x}(x,y)$$
Since you've been discussing when you know the number of variables of the function $f$, I'll try to clarify it :
If on your text, exercise or solution, you have used the following notation : 
$$f(x,y) \space \text{such that} \space f : D \to \mathbb R^2 \space\text{where} \space D \subset \mathbb R^2 $$
then that means that $f$ is strictly a $2$-variable function, since for example, writing $f$ in that way : 
$$f(x,y,0)$$
is wrong. Why ? Because the domain that $f$ is set is the set $D$ which is a subset of $\mathbb R^2$. This means that the element $(0,0,0)$ cannot be included in $D$ because it's set in $2$ dimensions.
If you are not given such a complete notation (which is impossible in books and exercises, since functions are almost always defined first of all as $f : A \to B$ and then as $f(x,\dots)$) then you may argue about the notation. Still, the most convenient and usual way of writing is the one mentioned above. If though, you're given such a notation as mentioned, then there is no question about the variables-dimensions of $f$.
A: The function is multivariate anyway, thus the symbol 
$$
\frac{\partial}{\partial x}f(x,y)
$$ is clearer than the other one.
