Can Differentials and Errors be Split in Calculus?

[KingBigness]

I asked a question a few weeks ago about 'splitting' the derivative. The thread can be found https://www.physicsforums.com/showthread.php?p=3581188#post3581188"

The answer to why it can not be split is because dx does not exist, it is simple a notation and not a fraction.

However, I just started Differentials and Errors, and the paper I read said.

The ratio of the two derivatives is actually the derivative of the function.

$f'(x)=\frac{dy}{dx}$

and the relationship between the two differentials can be given by

$dy=f'(x)dx$

Is this not 'splitting' the derivative?

Thank you

 

[lurflurf]

dx does exist and dy=f′(x)dx is fine

The actual definitions in use can very and sometimes confusion can arise. Often the implicit function theorem and its conditions are in play.

 

[vrmuth]

The answer to why it can not be split is because dx does not exist, it is simple a notation and not a fraction.The ratio of the two derivatives is actually the derivative of the function.
$f'(x)=\frac{dy}{dx}$

and the relationship between the two differentials can be given by
$dy=f'(x)dx$

Is this not 'splitting' the derivative?

I think you mean the ratio of two differentials , yes you are right , dy and dx are merely notations , and don't exist separately but when you write $dy=f'(x)dx$ , dy becomes function of variables x and dx , and meaningful when used together as in the case of dy/dx , also see the following link
http://en.wikipedia.org/wiki/Differential_of_a_function"

 

[lurflurf]

That is like saying 3/4 is merely a notation and 3 and 4 do not exist separately.
The advantage of differentials is when we have several variable we can understand how each changes with the others independent of the representation as differentials are invariant. dy/dx means the same thing if we have
y=f(x)
x=g(y)
h(x,y)=0
w(u(x,y),v(x,y))=0
and so on
it would be quite limiting to say dy/dx=f'(x) the end, if we change variables we will start from scratch.

 

[vrmuth]

That is like saying 3/4 is merely a notation and 3 and 4 do not exist separately.
The advantage of differentials is when we have several variable we can understand how each changes with the others independent of the representation as differentials are invariant. dy/dx means the same thing if we have
y=f(x)
x=g(y)
h(x,y)=0
w(u(x,y),v(x,y))=0
and so on
it would be quite limiting to say dy/dx=f'(x) the end, if we change variables we will start from scratch.

well said ! , But i don't say dy/dx is merely a notation , tell me what's dx ? ( without talking anything about dy)

 

[lurflurf]

Again different definitions of differential are used in different contexts so this is for the purpose of regular calculus
dx is just some variable for an arbitrary change in x
which is not the most interesting part
given some sufficiently well-behaved curve or relationship of variables we can define a tangent space that depends upon the point and the curve at that point, but in no way depends upon specific coordinate choices. The projection of the curve is a linear function of the differentials and approximates the curve well near the point.
thus
f(x+dx)-f(x) is some function of x and dx
and
f'(x)dx is some function of x and dx
they do not agree in general only approximately near x

This only works so well for total differentials of first order.