Why we can multiple dx on both side in df/dx equation

Question

I'm reading about Total Derivative when learning machine learning concept. I'm reading to this:

The thing I don't know is: As my knowledge, df/dt looks like a "notation" than a variable. But in this, I see that we can multiple by both side dt. It looks like dt is acting as variable. I don't get this point. Can explain for me, why can we do this. And what does the meaning of dtwhen it stands as single variable.

Thanks

Answer 1

The definition of "differential" has the equation

$$dy = f'(x) \; dx$$

or $$dy = \frac{dy}{dx} \; dx.$$

It looks like cancellation, but it isn't really. But it does say that whenever you have

$$\frac{dy}{dx} = \mbox{crud}$$

you can multiply by $dx$ to get

$$\frac{dy}{dx} \; dx= \mbox{crud} \; dx.$$

Then you can use the equation from the definition of differential to replace $\frac{dy}{dx} \; dx$ by $dy$. So it's replacement, not cancellation.

Edit: In the definition of "differential", $dx$ and $dy$ are new variables. The equation $dy = f'(x) \; dx$ shows what the relation ship is between those variables and $x$ (and sometimes $y$, too.) If you're given a specific point $(x,y)$ on the graph of $y=f(x)$, then you can think of the origin of the $dx-dy$-plane as being on that point, with the $dx$-axis parallel to the $x-axis$ and the $dy$-axis parallel to the $y$-axis. The $dx-dy$-plane is a little traveling coordinate system that goes along the function. The equation $dy = f'(x) \; dx$ is the equation of a line through the origin of that space.