Division by $dx$ in multi-variable calculus .... I am stuck on this doubt :
Suppose $f=f(x,y,z).$ Hence, $ df= \frac {\partial f}{\partial x}dx + \frac { \partial f}{\partial y}dy + \frac {\partial f}{\partial z}dz.$ 
Then, is the following equation correct :
$$\frac {df}{dx}=\frac {\partial f}{\partial x}+\frac {\partial f}{\partial y}\frac{dy}{dx} + \frac {\partial f}{\partial z}\frac{dz}{dx} \,\,\,\,(*)$$
The reasoning used in obtaining $(*)$ is : "dividing" the whole equation by $dx$. Normally, the $\large \frac {d}{dx}$ operator is used in single variable calculus where only single-variable functions are differentiated wrt $x$. But it does look a bit awkward (at least to me) when used in multi-variable calculus. Do the expressions $\large \frac {df}{dx}$, $\large \frac {dy}{dx}$ and $\large \frac {dz}{dx}$ even make any sense when used like this ?
I know that "division" by $\partial x$ can cause problems in multi-variable calculus. But what about "division" by $dx$. It works fine in single-variable calculus.
If $\large \frac {df}{dx}$ makes any sense, then does it mean the "total" rate of change of $f$ wrt $x$ if $y$ and $z$ are allowed to change ?
Summary : 
(1) Is division by $dx$ allowed in multi-variable calculus? 
(2) What does $\frac {df(x,y,z)}{dx}$ mean if answer to (1) is "yes" ? Does it mean anything if the answer to $(1)$ is "no" ?

If $$f=f(x,y,z)$$ where $x,y,z$ are independent variables, then you may divide $df$ by $dx$.
For example $$f(x,y,z)= xyz+x^2+y^2+z^2$$
$$df = (yz+2x)dx + (xz+2y)dy + (xy+2z)dz$$
$$\frac {df}{dx} = yz+2x + (xz+2y)\frac {dy}{dx} + (xy+2z)\frac {dz}{dx} =yz+2x $$ 
Which is  the same thing as $\frac {\partial{f}}{\partial {x}}$
