Shorthand notation for partial? If I am taking a regular derivative, and I want to show the process in detail, I'll do something of the sort $f'(x) = g'(x) + h'(x) - l'(x) ..... $, etc, using that "prime" notation. 
However, what if I wanted to take the partial derivative? Would I still use the prime notation, as long as it's clear what we are treating as a constant? Or is there some other notation for this? 
 A: Prime notation for multivariate functions is very confusing.
Two common notations for partial derivatives are$$\frac{\partial}{\partial x} f(x,y) = f_x(x,y).$$
Similarly $$\frac{\partial}{\partial y} f(x,y) = f_y(x,y).$$
A: If you define that the "long"-hand notation is:
$\dfrac{\partial^n f}{\partial x_1 \partial x_2 \dots \partial x_n}$
then a shorthand can be $\partial_{x_1\,x_2\,\dots\,x_n} f$, where you deduce the degree of the derivative by the number of footers of "$\partial$".
A: You have a few options. As noted by others $\frac{\partial f}{\partial x}$ and $f_x$ both commonly express the derivative of $f$ with respect to $x$.
Another option is $D_x f$. To explain, $D$ is commonly notation for the differential operator, the function that maps a function to its derivative. For instance, $D(f) = f'$. Then adding the subscript indicates taking a partial derivative, so $D_x(f)=\frac{\partial f}{\partial x}$. Writing it out we can often omit the parentheses to make the notation even lighter, so we get $D_x f$ denoting the partial derivative of $f$ with respect to $x$.
IMO, the notation $\frac{\partial f}{\partial x}$ is universally recognized but cluttered. The notation $f_x$ is very compact but isn't always obvious. So I like the notation $D_x f$, it reinforces the idea that we aren't referring to $f$ itself but its related partial derivative.
