What are the different interpretations of differentials in calculus?

[noslen]

Notation of derivative?? help

Where does dy/dx come from I know what it means but is this right


y=x^2+2x

dy=2x*dx+2*dx

dy/dx=(2x*dx+2*dx)/dx

dy/dx=2x+2

Im not sure if this is right can someone please explian thanks

 

[Gaz031]

Where does dy/dx come from I know what it means but is this right


y=x^2+2x

dy=2x*dx+2*dx

dy/dx=(2x*dx+2*dx)/dx

dy/dx=2x+2

Im not sure if this is right can someone please explian thanks

if $$y=f(x)$$:
$$\frac{dy}{dx}=\lim_{h \rightarrow 0} \left(\frac{f(x+h)-f(x)}{h}\right)$$

 

[dextercioby]

Differentiating a function & dividing through the differential of the variable = Computing the derivative.

Daniel.

 

[noslen]

So is this correct Daniel?

 

[dextercioby]

Yes, it is, but not mathematically rigurous to divide through "dx", just as if it was a number. However, it leads to correct results (just like separation of variables in a I-st order ODE).

Daniel.

 

[Zurtex]

Yes, it is, but not mathematically rigurous to divide through "dx", just as if it was a number. However, it leads to correct results (just like separation of variables in a I-st order ODE).

Daniel.

In my class we were shown rigourously how to do it and if we ever used something like "dy = 2x dx" then we were told only ever to have it in quotes like so to show it wasn't really true.

We were shown how to deal with all situations while leaving it in its rigorous form and what all the steps in between were when we made things like substitutions for integration. I actually found a bit of a struggle on this forum to deal with reading things like the above statements. Guess it's how you teach it, but I've found people at university to get very confused about issues generally around 'infinitesimals' when the lax notation is used.

 

[Hurkyl]

The really confusing thing is that one may eventually learn that "dy = 2x dx" is a perfectly rigorous notation.

Specifically, it's the statement that the two differential forms "dy" and "2x dx" are equal on the curve given by the equation y = x^2. (A differential form is, loosely speaking, something you can integrate)

 

[lurflurf]

The really confusing thing is that one may eventually learn that "dy = 2x dx" is a perfectly rigorous notation.

Specifically, it's the statement that the two differential forms "dy" and "2x dx" are equal on the curve given by the equation y = x^2. (A differential form is, loosely speaking, something you can integrate)

Yes there are several ways to think of something like
y=x^2
hence
dy=2x dx
1) as an "abuse of notation" which gives correct answers but is not logically sound.
2) where dx means difference instead of derivative and the equals is approximate and holds for small dx
dx=(f(x+dx)-f(x))=f'(x)dx+f''(x)dx^2/2+f'''(x)dx^3/6+...~f'(x)dx
3) where 2x dx and dy are differential forms
then dy is defined as dy=f'(x)dx but the dx is not really the differential of x in a sense it is another variable. That is it is just some number not an infinitessimal.
4) in nonstandard analysis the tools for working with infinitessimal quantities is developed. In this frame work dx and dy are actually infinitessimal and their quotient if defined and f'(x)=dy/dx is really a fraction of infinitessimals.

It is quite confusing how a student in intro to calc should interpet differentials.