A question on the difference between $\frac{\partial f}{\partial x}$ and $f_x$. I have a question related to partial differential equations:
Say we have $f(x,y,g(x,z))$. Is $f_x\neq \frac{\partial f}{\partial x}$?
By what I have read, $f_x)$ is the derivative of $f$ assuming $g(x,z)$ and $y$ to be constant. 
 A: They are the same whatever $f$ is. It is just a symbol to the first partial derivative of $f$ with respect to $x$. When you try to get this partial derivative, you have to assume that the independent variables be constant. In this case $g(x,y)$ depends on $x,y$, so you can not assume it as constant.
A: Personally, I would use $f_x=\frac{\partial f}{\partial x}$. However, perhaps the notation below would be helpful:
$$ \frac{df}{dx} = D_1f+(D_3f)g_x $$
Or, if you like, view $x$ and $y$ as functions of $t$ so
$$ \frac{d}{dt}(f(x,y,g(x,y)) = \frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}+\frac{\partial f}{\partial z}\frac{dg}{dt} $$
But, setting $t=x$ this yields:
$$ \frac{d}{dx}(f(x,y,g(x,y)) = \frac{\partial f}{\partial x}+\frac{\partial f}{\partial z}\frac{dg}{dx} $$
and, as $\frac{dg}{dx}  = \frac{\partial g}{\partial x}\frac{dx}{dx}+ \frac{\partial g}{\partial y}\frac{dy}{dx} = \frac{\partial g}{\partial x}$
$$ \frac{df}{dx} = \frac{\partial f}{\partial x}+\frac{\partial f}{\partial z}\frac{\partial g}{\partial x} $$
Moral of story: don't use the coordinate as a parameter unless you are prepared to deal with this ambiguity.
