TITLE: Notation regarding different derivatives
QUESTION [11 upvotes]: I am currently reading up on partial derivatives and differentials in general. And there are a few points that seem unlcear to me (notation-wise). 

For example, if $f:\mathbb R\to\mathbb R,x\mapsto f(x)$ is a function, then is the following notation correct?
$$\frac{d}{dx}f(x)=\frac{df}{dx}(x)=\frac{\partial}{\partial x}f(x)=\frac{\partial f}{\partial x}(x)=f'(x)$$
Now, in one of my lectures we wrote
$$g(x):=\frac{\partial \log (f(x))}{\partial x}=\log'f(x)\cdot f'(x)=f'(x)/f(x)$$
The part about differentiating the function  is clear, using the normal chain rule $(h\circ f)=(h'\circ f)\cdot f'$. What  confuses me a bit is the notation since $x$ seems to have more than one meaning. Would it be "more" correct to write
$$g(x):=\frac{\partial \log (f(y))}{\partial y}\Bigg|_{y=x}$$
as in we differentiate w.r.t. $y$ and then plug in $x$ for $y$? Also, in this particular case, could we replace $\partial$ by $d$, or would this lead to different implications?
And lastly, what does $df$ or $dy$ even mean? I read that the differential of $y=f(x)$ is defined to be $dy=f'(x)dx$. But how can we interpret this formula? This becomes specifically confusing when I look at the differential chain rule
$$\frac{dh}{df}=\frac{dh}{dg}\cdot\frac{dg}{df}$$
What I find strange is that one seems to be using $g$ as a function as well as a variable. How does this work exactly?

REPLY [8 votes]: Let $x\in \Bbb R$. 

I find the red notations below pedagogically problematic $$\color{red}{\frac{d}{dx}f(x)}=\frac{df}{dx}(x)=\color{red}{\frac{\partial}{\partial x}f(x)}=\frac{\partial f}{\partial x}(x)=f'(x),$$ but they are OK just as long as you don't look at $\color{red}{f(x)}$ as $f$ evaluated at $x$, but rather look at $\dfrac{d}{dx}f(x)$ and $\dfrac{\partial}{\partial x}f(x)$ as the derivative of $f$ evaluated at $x$.It should be mentioned that $d$ is usually used when it is a function of one variable and $\partial$ when it is a function of more than one variable.
You're absolutely right. The notation $\dfrac{\partial \log (\varphi(x))}{\partial x}$ is simply wrong. Just look at the notations on item 1. Does it look like any of them? No. Even if you add the notation $\dfrac{\partial f(x)}{\partial x}$ to the list (and this is also a notation I deem bad) and interpret it as the derivative of $f$ at $x$, it still doesn't concurr with $\dfrac{\partial \log (\varphi(x))}{\partial x}$. What would your $f$ be here? If you're consistent, it would be $f=\log$, but then $\dfrac{\partial \log (\varphi(x))}{\partial x}$ would be, according to our interpretation, the derivative of $\log$ evaluated at $\varphi(x)$, i.e., $\dfrac{1}{\varphi(x)}$. What it is intended is $f=\log \circ \varphi$, but then the notation should be $\dfrac{\partial (\log \circ \varphi)(x)}{\partial x}$ which something completly different. With the notation $\dfrac{\partial (\log \circ \varphi)(x)}{\partial x}$ you do get $\dfrac{\partial (\log \circ \varphi)(x)}{\partial x}=\dfrac{\varphi'(x)}{\varphi(x)}$ as wanted.
Indeed it would be more correct to write $\dfrac{\partial \log (f(y))}{\partial y}\Bigg|_{y=x}$, but not entirely correct because, once again, the notation isn't  on the list you gave on 1. After you add notation $\dfrac{\partial f(x)}{\partial x}$ and give it a proper interpretation, then yes, it is correct.